I wonder whether maths really can be self-taught for more than about two generations. It has been known since the beginning of maths that it must be taught by teachers who understand the ideas to students who challenge the descriptions.
Yes, Math can be self-taught. I am a living example. I went through school, up to 10th grade (junior high school) with only getting minimum passing grades. I was worse at everything except History. That was the only thing that I liked. And even at History I was getting like medium grades, not the highest marks.
Then in the summer vacation between 10th grade and 11th grade I got hit with a hobby of Electronics. I read by mistake an old book, from 60's, that was in my father's book collection, of how to create a radio for short waves that was powered by a potato (the vegetable, not the sarcasm!). And I went down through that rabbit hole. I started to learn electronic components and what they do. That required Physics, which I knew nothing about it - so I started to learn Physics as well. Well, have you ever encountered Physics to be, in practice, applicable without Math? So I started to learn Math all by myself. Between ages of 16 to 18, while I was as Senior in high school I recovered all previous 10 years when I slacked worse than a worm. That allowed me to go to University where I got hit by another hobby - programming. Both these hobbies still stay with me to this day.
So yeah, I would argue not only is possible to learn Math all by yourself, but I'd say that doing so you actually learn it even better then when is forced upon you, even if you're good at it.
This is a good point, but I think it is relevant to ask whether most people learning math in school have the experience you describe. I certainly did not. My math education consisted purely of recitation of things I didn't fully understand, and I got very good grades all the way through.
When you say self-taught do you mean even without using a really good text? Or do you mean that it actually can't be taught without a teacher? If you are saying that math can't be taught without a good teacher I don't really see how that is "known".
Even if it requires a good teacher, does society benefit from forcing it on individuals at a time they don't want to learn it? Or would we be better served by making good teachers available to those who want to learn it, at the time they want to learn it?
I'd -love- to go through some of my college level math classes; both the ones I took, and the ones i didn't need to take. They weren't especially relevant then. They're slightly more now, but more than that, I'm -interested- now. But self-learning takes too much of an investment of time (when I run into something I don't know I have to research -that-, and it becomes this infinite process of diving down rabbit holes, rather than having someone who has gone before who can give me a sufficient answer to unblock me on my original question).
If anything, math is probably one of the easiest to self teach. The basics (which goes all the way up early university level) hasn't really changed in 100s of years, and learning resources are readily available, and you don't need any special equipment.
