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CronoDAS comments on Not for the Sake of Pleasure Alone - Less Wrong
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If you include the time spent in elementary school, high school, and college, most people with a Ph.D in math have spent many, many years studying math...
Also, generally "prodigy" means that, as a child, one was far beyond one's age group. If you're learning algebra at 8 and calculus at 11, you're a prodigy... even if you don't yet know any math beyond the high school level.
That doesn't feel sufficient to me. I usually interpret the word to imply achieving high levels of status while still a child (for example, winning national competitions, touring internationally as a perfomer, etc.). Merely learning stuff won't do that.