Math is the purest form of art humans can produce.I'd say one can learn a fair bit of it on his own and be relatively knowledgeable,but..There are many many things in math that are not just mind boggling,but also so difficult that unless you can singlehandedly crack problems that haven't been solved for centuries, you'd need a support from someone more senior. Those more seniors most likely will be found at good universities teaching and researching the subject.