Define 'good'. I've seen people in their early 20s go from one branch (say, combinatorics) to another branch (say, algebraic geometry) without taking any formal courses [one could argue this is equivalent to going from being a great neurosurgeon then 6 months later publishing papers at the forefront of pancreatitis research]. They end up out performing post-docs who've spent a decade solely in that field, often within months. In my mind that's just as impressive as the Ramanujans and all (who was 'discovered by Hardy' but still had quite a base line of formal training prior to being invited to England).
Mathematics has developed so extensively that there's going to be a base corpus of knowledge one generally has to acquire before they're at the caliber of being able to publish. Take a tenured topologist and place him in at a conference where people are presenting their findings in an entirely different field and he'll more often than not struggle to keep up. Combined with the fact that higher level education is a lot easier to attain than it was say, 50 years ago (and definitely 100 years ago) allowing a larger percent of population who those who want to enter into pure mathematics to do so, it's fairly understandable why you won't see many 'good' self-taught mathematicians. (Though, I argue that recluses like Perelman who go off the grid for years at a time after being inspired by something like Ricci flow to solve Poincare are the analogues of yesteryears self-taught mathematicians, as they are no longer collaborating with institutional academia.)
Going from combinatorics to algebraic geometry with formal training in the former but not the latter doesn't count in my opinion. Having formal training in pure math makes one more able to self learn other branches. I have a hard time believing that one can go from undergraduate calculus/linear algebra to algebraic geometry in anything more frequently than than extremely rarely.
The first hump is understanding a branch of mathematics. Once that is done one is likely to be able to self learn other branches. Can that first hump be done except in rare cases? I don't think so.
No one is asserting that someone taking 100 level calculus is going to self-teach => publishing in algebraic geometry. I agree that your first hump correctly describes the first barrier. I assert that a lateral movement in 'sufficiently different' subsets of mathematics will provide an additional hump within the space of knowledge, quantified by a magnitude of eh maybe > 2/3s minimum. Going from comb. to alg. geo. is amazingly hard; hard enough, I suggest, that it's the equivalent of modern day 'self-teaching', given the large corpus of knowledge one has to attain[1]. I assert it'd take at least ~2 years for one to move from combinatorics to an understanding of say, Sheaf Theory circa 1950 (up to Serre) unless you're one of those rare guys like Tao who can jump from field to field (ugh pun not intended).
[1]http://matt.might.net/articles/phd-school-in-pictures/ Getting published in the J. of Top. is being on the arc (perhaps, even deforming the disc in R^2 heh heh). Being able to move laterally s.t. your findings substantial enough they are accepted into the J. of Alg. Geo. requires a requires such an absurdly large lateral movement along the arc that it's a feat analogous in difficulty to self-teaching oneself up until say the 1920s.
Yes they are. this whole thread is about mathematics being accessible to people outside academia. the greatgrandparent poster posted that they could follow mathematics despite having skipped university in a thread about mathematical maturity and being accessibility outside of formal training. someone pointed out that you could cross disciplines but this isn't what we were talking about as this builds on top of a classical training in university.
The person I responded to seems to think that mathematics is not difficult to engage with outside of academia. You seem to be under the impression that even making a lateral move post professional training is amazingly hard. It appears that you agree with my first post that grasping second year graduate level mathematics does require formal training for all but a very few.
Mathematics has developed so extensively that there's going to be a base corpus of knowledge one generally has to acquire before they're at the caliber of being able to publish. Take a tenured topologist and place him in at a conference where people are presenting their findings in an entirely different field and he'll more often than not struggle to keep up. Combined with the fact that higher level education is a lot easier to attain than it was say, 50 years ago (and definitely 100 years ago) allowing a larger percent of population who those who want to enter into pure mathematics to do so, it's fairly understandable why you won't see many 'good' self-taught mathematicians. (Though, I argue that recluses like Perelman who go off the grid for years at a time after being inspired by something like Ricci flow to solve Poincare are the analogues of yesteryears self-taught mathematicians, as they are no longer collaborating with institutional academia.)