I'm surprised you use Mathematics as an example of a field that's hard to self teach. I actually think Mathematics is one of the easiest fields to self-teach in - you really only need books / video lectures, some paper and pencils, and your brain. It's even less demanding than programming in that sense!
There are also several famous self-taught Mathematicians.
Programming is easy to self-teach because you can immediately verify whether you've got it right -- compile and run, see if it works.
Mathematics is harder to verify, though easier than (say) physics because you can test yourself by proving theorems. However, your proofs might be wrong because you're under some misconception. And, as the other response said, it requires an incredible amount of discipline.
Ramanujan was probably the last great self-taught mathematician, and I'm sure most would say he could have been a helluva lot greater if only he'd been sent off to Oxford at sixteen instead of twenty-seven.
The fundamental things being taught to math majors are how to distinguish between valid and invalid reasoning, and how to produce arguments (proofs) that contain only valid reasoning.
For better or worse, the way these skills are taught is to pick a couple of subjects (like analysis, algebra, or number theory) and demand that students produce dozens of proofs, which are then mercilessly picked apart by professors and grad students. That negative reinforcement is combined with lectures that (should) consist entirely of good examples of mathematical reasoning, at that's an Education.
(The fact that students may also pick up a few facts about analysis or algebra is just a bonus, it's not the main point.)
You can certainly get the good examples from books and video lectures, but not the experience of having your arguments criticized by experts. This part is crucial -- it's just as easy to miss the flaws in your own proofs, especially your early attempts, as it is to miss the flaws in your own programs. (Worse, it's perfectly possible to reach a conclusion that's correct through an invalid argument. You're wrong, but you'll never be able to find evidence against your conclusion.) Feedback from experts is essential.
You also need lots and lots of self-discipline. Of course, that is true for serious study in any field, but unlike many fields no one would call a mathematician someone who wasn't serious about it.
"Which is ye surest character of a true Mathematical Genius, learned these of his own inclination & by his owne industry without a teacher."
— Isaac Newton
In many respects, math has a totally unfair advantage over every other subject you could possibly study in college because it can be learned almost without needing any other people. Well, living people that is, as you do need the books of long deceased thinkers. (I'd say this also applies to philosophy, and hardly anything else). You just need to be willing to spend enormous amounts of time alone in contemplation reading.
The sole cause of man's unhappiness is that he does not know how to stay quietly in his room.
— Blaise Pascal
There's also a very limited social component layer such as you would find in medicine, law, politics, economics, et al. To become a doctor, or any kind of "professional", sure, it requires a massive social apparatus on top of a high degree of actual learning. You have to have connections, know the right people, say the right and popular things to them, be supported and credentialed by the leadership and their agenda, and basically imposture one's self into mimicking whatever behaviors and traits are specific to and dominant in whatever professional field you're studying. Eventually your identity morphs into this set of learned behaviors, and it's almost entirely built upon your position in some type of social hierarchy.
Not so with math. You either know some theorem and the proof, or you don't. How much you know and how advanced you are is entirely dependent upon what you have worked through on your own. What new stuff you can create only comes from what old stuff you already know. None of this has anything to do with interacting with people. Sure, you can benefit by talking to other mathematicians, attending lectures, conferences, teaching others, etc, but at the end of the day, at some point, you still have to sit down and learn the material by yourself. And if you don't do that, it doesn't matter how much of a social butterfly you are—you can't be a mathematician.
No man is an island, but I imagine a Robinson Crusoe type figure would have no such problems becoming a mathematician on a desert island, given enough math books(and some kind of Internet connection).
Well, it was a lot easier to be self-taught in Newton's day... there wasn't that much to learn! You could read Euclid and you're be mostly done. Newton alone probably made learning mathematics at least three times harder during his career.
Anyway, if you're trying to make me regret choosing mathematics as one of my examples, y'all have won. I see no molecular biologists have popped up to advocate for self-taught molecular biology.
I've read that the last Universal mathematicians we've had were Chebyshev and Poincare, who could claim to know "all of math", and who were active in research in nearly every sub-field. It's been over 50 years since their time, and much like the fabled Renaissance Man, that frontier is now closed.
There are also several famous self-taught Mathematicians.