[gjulianm]

As a mathematician, and seeing what's on the article (and with no intention to downplay your achievements, which are impressive), that's not what I think of when I hear "advanced mathematics". Vectorial calculus and differential equations (ordinary, not partial) are basic courses in math degrees. For the things that the article explains, such as topology, group/ring theory, measure theory, functional analysis, etc (which are still nothing fancy that doesn't get reviewed in a degree, so not yet "advanced"), I think that self-learning is almost impossible unless you're near to a Terence Tao-level genius.

Here I talk from experience. I remember reading books on some of these subjects and understanding few things, without really getting a grasp of what they're talking about. A lot of times, the problem is that you don't know what is missing in your knowledge. You need a clear roadmap, you need relationships, you need to solve a lot of questions, you need to do exams and, most importantly, you need to test your knowledge. I cannot even count how many time I thought I understood some theorem only to do some exercise and see that I had absolutely no idea. Sometimes you notice yourself, sometimes you do it so bad that you don't even notice it is incorrect.

And, for these subjects, the material on the Internet starts to diminish and be less accessible (more oriented to professional mathematicians than to learners). Khan Academy does not have advanced courses, the definitions on Wolfram or Wikipedia are only useful if you have already a grasp of the subject (see for example https://en.wikipedia.org/wiki/Measure_(mathematics)#Definiti... - What is important? What are the critical aspects? Which are the subtle parts of the definition that you must read carefully?) and in Youtube you may find lectures, but usually they're like the books: you will be lucky if it's not a succession of theorems and definitions, and you still lack the possibility of checking and testing your knowledge.

So, while some parts of math can be learned independently, I don't think that advanced mathematics can be done. Myself, only after 5 years of mathematics I'm somehow comfortable to study subjects by myself, and it's still hard.

[kragen]

My grasp of group theory, measure theory, and functional analysis are fairly weak, so maybe I'm not the best person to comment on this, but I think the problem may be that you were overoptimistic when you attempted to read those books. Usually when I read books on subjects I don't understand, I don't understand the book the first time I read it. Reading several different books on the subject helps. This requires a lot of persistence and tolerance for frustration. But that's true when you take a class, too!

As you say, though, you need to solve a lot of questions (which I interpret to mean "do a lot of exercises" or "do a lot of problem sets") to understand something. Reading a textbook without doing exercises is minimally useful, although it can help with the "roadmap"/"relationships" thing. Wikipedia is usually a pretty good roadmap, too, although it varies by field.

But you can also read textbooks and do exercises. This depends on the existence of, and access to, sufficient textbooks and exercises, but Library Genesis has recently extended that kind of access to most of the world. Taking functional analysis as your example, the 1978 edition of Kreyszig is on there, and it averages about two exercises per page, and has answers to the odd-numbered ones in the back. This quantity of exercises seems like it would probably be overkill if you were taking a class in functional analysis and could therefore visit the professor during office hours to clear up your doubts, but it seems like it would be ideal for self-study. And if two exercises per page isn't enough, you can get more exercises out of a different textbook, like Maddox (1970 edition on libgen) and Conway (first and second editions on libgen). You can find textbooks on scholar.google.com by searching for the names of general topics and then looking for "related articles" with thousands of citations, because for some reason people like to cite their textbooks.

Unless you can find a desperate adjunct math faculty member looking to make some extra bucks on the side or something, it's true that comparing your answers to the exercises to those given isn't as good as having a TA actually correct your homework. But it's usually good enough.

(Of course you should only download these books if that wouldn't be a violation of copyright, for example, if their authors granted libgen permission to redistribute them or you live in a country not party to the Berne Convention.)

Progress will be slow. But I think the key thing here is to start with low expectations: expect that you'll manage to read about 15 pages a week and understand half of them. I don't think you have to be a Terence-Tao-level genius.

[wnewman]

(responding not to what is in the article, but only to your comment on how difficult it is to study what is more nearly "advanced mathematics")

I got 800 on the 1980s-era math SATs, came in third in the Portland OR area in a math contest in high school, and did OK at Caltech (not in a math major), but I'm no Terry Tao, and I very much doubt I'd've been anything very special in a good math undergrad program. Some years after graduation, I found it challenging but doable to get my mind around a fair fraction of an abstract-algebra-for-math-sophomores textbook, including a reasonable amount of group theory (enough to formalize a significant amount of the proof of Solow theorem as an exercise in HOL Light, and also various parts of the basics of how to get to the famous result on impossibility of a closed-form solution for roots of a quintic).

From what I've seen of real analysis and measure theory (a real analysis course in grad school motivated by practical path integral Monte Carlo calculations, plus various skimming of texts over the years), it'd be similarly manageable to self-learn it.

One problem is that some math topics tend to be poorly treated for self-learning, not because they are insanely difficult but because the author seems never to have stepped back and carefully figured out how to express what is going on in a precise self-contained way, just relying (I guess) on a lot of informal backup from a teaching assistant explaining things behind the scenes. On a small scale, some important bit of notation or terminology can be left undefined, which is usually not too bad with modern search engines but was a potential PITA before that. On a larger scale, I found the treatment of basic category theory in several introductory abstract algebra texts seemed prone to this kind of sloppiness, not taking adequate care to ground definitions and concepts in terms of definitions and concepts that a self-studying student could be expected to know, and that's harder to solve with a search engine, tending to lead into a tangle of much more category theory and abstraction than one needs to know for the purpose at hand. My impression is that mathematicians are worse at this than they need to be, in particular worse than physicists: various things in quantum mechanics seem as nontrivial and slippery as category theory to me, but the physicists seem to be better at introducing it and grounding it. (Admittedly, though, physicists can ground it in a series of motivating concrete experiments, which is an aid to keeping their arguments straight which the mathematicians have to do without.)

I have been much more motivated to study CS-related and machine-learning-related stuff than pure math, and I have been about as motivated to self-study other things (like electronics and history) as pure math, so I have probably put only a handful of man-months into math over the years. If I had put several man-years into it, it seems possible that I could have made progress at a useful fraction of the speed of progress I'd expect from taking college math courses in the usual way.

I think it would be particularly manageable to get up to speed on particular applications by self-study: not an overview of group theory in the abstract, but learning the part of group theory needed to understand the famous proof about roots of the quintic, or something hairier like (some manageable-size fraction of) the proof of the classification of finite simple groups. Still not easy, likely a level harder than teaching oneself programming, but not an incredible intellectual tour de force.

"Myself, only after 5 years of mathematics I'm somehow comfortable to study subjects by myself, and it's still hard."

Serious math seems to be reasonably difficult, self-study or not. Even people taking college courses in the ordinary way are seldom able to coast, right?

[sudoscript]

As someone self-studying measure theory right now, I completely agree on the quality of math textbooks for more esoteric subjects. It's like the authors expect the books to only be used in conjunction with TAs or classes.

Any advice on how to use those textbooks the best way?