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If Euler, Gauss, Maxwell, even Ramanujan, Galois, etc., needed to spin up on foundational knowledge, then so do you. Yes, these people all produced plenty of original research at early ages, but they didn’t skip the foundations.
Even Ramanujan self-studied. He didn’t just sit there and think up math with no prerequisites in place. In his teens he reportedly worked through Carr’s A Synopsis of Elementary Results in Pure and Applied Mathematics, consisting of 5,000 theorems.
And Galois? Same thing. Galois had a multi-year period of mathematical upskilling before making creative contributions. By his mid teens he had accumulated several years of formal schooling that included math. Then he caught the math bug, became obsessed with Legendre’s geometry textbook, worked through it, and continued working through key research papers for a couple years afterwards. And THEN he started producing his own creative contributions.
(Keep in mind there were also far fewer prerequisites needed to make novel contributions to math back then, so a person could fill up their prerequisites to cutting-edge math in a shorter time. Additionally individuals were presumably highly cognitively advantaged with outsized learning rate and generalization ability, which would allow them to move faster through a curriculum, infer more knowledge beyond it, and infer more knowledge from self-directed experimentation. Basically, across all math learners, these individuals would have been some of the most likely to have their prerequisites in place.)
In general, when it looks like someone progressed so fast they “must” have taken a shortcut, what really happened is they speed-ran the foundations. Either that or you’re overestimating their actual ability (likely because they’re exploiting signaling to trick you).
How can people speed-run the foundations? By way of a more efficient training environment, advantageous individual differences leading to more rapid skill acquisition, or by allocating way more of their time into training than is typical. Elite performers typically emerge from a combination of all three of those things.
Take Euler for example.
- Advantageous training environment? Check. Euler was taught by his father, who had studied under Jacob Bernoulli, and Euler enrolled in the University of Basel at 13 years old, where he studied under Johann Bernoulli and self-studied more advanced textbooks under Johann’s guidance.
- Allocating way more of their time into training than is typical? Check. I mean, geez, what 13-year-old is taking university courses and self-studying even more math textbooks on top of that?
- Advantageous individual differences leading to more rapid skill acquisition? I don’t think there’s any formal record, but you’ll see Euler on pretty much any “highest estimated IQ of all time” list. It seems pretty safe to conclude he was running on some exceptional cognitive machinery.
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