Perhaps unrelated, but I wonder who these mentors were in the case of people like da Vinci, Socrates, Archimedes, Newton et al. They must have had some adults who guided them toward what they're known for. It seems absurd to believe that they just happened to grow up like that. It's as if everybody today, once you skim the "Early Life" section, it turns out had something fantastic, like a mother who was a Fields medalist, or uncle who invented this or that. The more that I see, the more I'm of the opinion that "genius" is simply:
1. effort, usually from youth, that nobody knows about so it appears to be innate
2. the effort is motivated and guided by some mentor(s), usually people with serious qualifications, like your Fields medalist uncle deciding to take you under his wing, after you said "math is fun :D" one time at 7 years old when he told you his job was "to do math :)" upon you asking him as children like to do, and turn you into a Fields-winning adult
This reminds me of those people who pretend they're a genius because they can guess the day of the week if you give them a date, when the reality is that anybody can learn to do that because it's just an algorithm[1] that you can calculate in your head and practice to the point that you come off as if you have a photographic memory or something.
As well as mentors, we should consider the environment in which genius arose, their cultural context and influences, the zeitgeist, intellectual atmosphere, schools, parents, friends, colleagues, the books they read.
I think we overvalue the uniqueness of the individual in this hero worship of the lonely genius, as if a flower is independent of the earth from which it grows.
But then again, it's true that there are exceptional stars, singular phenomena that cannot be explained by the sum of its parts. I suppose that leap, the surprising distance between what was given and what the individual made of it, is what we call genius, talent, luck or hard work.
While upbringing ("nurture") is clearly a critical component to success in any specialized field, there is undeniably a "nature" component as well, and I doubt that the best nurture could overcome lack of natural talent anymore than natural talent could overcome lack of nurture. To be the best you need a lot of both.
It's more obvious in athletics, where there is clearly natural talent, but the equalizer is the support network of parents funding practice, driving to games, getting into competitive leagues, etc. I don't see why the same wouldn't apply to intellectual talents as well, even if the differences in natural talent are less obvious than in athletics.
1. effort, usually from youth, that nobody knows about so it appears to be innate
2. the effort is motivated and guided by some mentor(s), usually people with serious qualifications, like your Fields medalist uncle deciding to take you under his wing, after you said "math is fun :D" one time at 7 years old when he told you his job was "to do math :)" upon you asking him as children like to do, and turn you into a Fields-winning adult
This reminds me of those people who pretend they're a genius because they can guess the day of the week if you give them a date, when the reality is that anybody can learn to do that because it's just an algorithm[1] that you can calculate in your head and practice to the point that you come off as if you have a photographic memory or something.
[1] https://en.wikipedia.org/wiki/Doomsday_rule