[gnulinux]

I strongly disagree actually. I think mathematics is the best field to self-study. There is nothing in mathematics that cannot be explained on the paper. There are many textbooks that are very good that in most universities professors won't be that good anyway. I went to UC Berkeley to study mathematics (ended up studying CS though) which is supposed to be a top department, but most of my textbooks were better teachers than my professors. I still prefer reading textbook to people explaining me math. I don't even think I understand math when people explain me. I need to first teach it to myself. Then occasionally people offering different perspectives is very beneficial, which, again, can be done on paper.

I don't think you need a PhD level educator. You need mathematical maturity. Mathematics follows a very specific logical structure that needs you to shift the way you think. Human brain simply doesn't work the way mathematics needs it to work. But this is a constant time overhead. Once you understand how to approach mathematical problems, I can't see why you cannot learn everything from a textbook.

[jack_h]

As someone who did study both math and CS at university I disagree. I think there are numerous courses that can more easily be self-taught[1], mostly what one would encounter in their first ~2 years in a math degree. After that things get conceptually a lot more difficult.

For me personally I didn't really need an instructor for most of my calculus courses, or ordinary differential equations, or most of the linear algebra stuff. It was a bit more difficult around real/complex analysis, non-linear dynamics, and courses of that nature. The classes that taught me the value of having an instructor were abstract algebra and topology. Those were such a massive shift away from what I had perceived math to be that an instructor being able to impart intuition, correct my own incomplete or incorrect assumptions, and generally just help guide me to a different mode of thinking was invaluable.

The problem with books/texts in this instance is they are not reactive, they have no idea what you're thinking and can't steer you in the right direction. Worse is that as the person trying to learn the subject matter you don't know where to look to get on the right track and correct your own assumptions because you don't know enough yet.

Now I'm not saying you need an instructor per se, but having some place to ask questions where someone far more knowledgeable than you can help might be a good substitute. I'm sure there are some websites like this, although I don't know of any since I graduated a long time ago.

[1]This does somewhat depend on a person's skillset going into this.

[asdffdsa]

The reality is that having a great mentor is a privilege, not a given. I don't think anyone is arguing that having a genius and brilliant teacher wouldn't help, rather that it is possible to reach an "advanced" (grad/undergrad level) understanding of mathematics without the luxury of having someone who is far more knowledgeable to turn to for help.

[gnulinux]

Precisely. GP here. I didn't argue against having a great mentor. If you go to a great university you clearly have an advantage. I think it's perfectly doable to teach yourself mathematics if you study intense enough eith correct tools.

[drablyechoes]

I think a big part of "mathematical maturity" is knowing how important it is to have someone else look over your proofs.

Of course you can learn a ton on your own by reading and working exercises and doing research, but there is no substitute for collaboration.

Mathematics is inherently a social activity, even if the bulk of it can (counter-intuitively) be done in relative solitude.

[decasteve]

I had the same experience. I tried to self study Math. I hit a wall where it became inefficient at best and beating my head against the wall for the rest. I couldn’t do it alone past material normally covered in the first two years of university.

I eventually did get to university and get a Math degree. It was much more rewarding and fun to do it with professors and other students.