It's really hard to say Ramanujan was a genius "without formal training." He didn't have the formal training of an English mathematician at Cambridge, but he had access to tutor and some material. He absolutely obsessed over what were essentially mathematical reference manuals for years in a way that was probably central to his extreme skill in symbol manipulation. It was a form of training that almost nobody would voluntarily expose themselves to.

It’s funny that you bring up Van Gogh, because he was as far from the archetypal genius as you can imagine. He started late, and he produced a lot of quite crappy paintings for many years before he produced his great works. And not least, he had a family that supported him so he didn’t have to waste his time with making a living.

Genius certainly exists, but it is often confused with the application of years of directed effort.

From the outside you might not be able to see the difference, but perhaps inside the discipline you can see the difference.

Often one might think that results are the indicator. But I don't Crick and Watson were genii (prefer to sound of geniuses) just hard workers but when the payoff of their hard work came out you would probably be excused for thinking they possessed that special gift.