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Epictetus comments on Innate Mathematical Ability - Less Wrong
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As Carl Linderholm pointed out, pattern-matching questions more properly belong to the field of parapsychology--he restricted his discussion to guessing the next number in a sequence, but the result can be readily generalized.
Satire aside, it seems to me that these Raven matrices get a lot easier to figure out once you've seen a few. At first glance I couldn't make heads or tails of the one you provided, but I went and took an online Raven matrix test and afterward that one seemed straightforward enough (in the sense that I quickly found a rule that was consistent with the rest of the matrix and produced one of the possible options). Presumably the easier ones familiarized me with the sorts of patterns the examiners were wont to use and reuse.
This reminds me of the Grothendieck quote from the previous article: "Yet it is not these gifts, nor the most determined ambition combined with irresistible will-power, that enables one to surmount the "invisible yet formidable boundaries " that encircle our universe." Both Grothendieck and Tao appear to discount pure intellect in favor of something less tangible when it comes to doing truly great mathematics. It's possible that they happened to encounter some exceptionally intelligent mathematicians who never managed to produce exceptional mathematics. On the other hand, it would be worth asking how many (if any) great mathematicians had high but non-exceptional intelligence.
Yes, Raven's problems do get easier when you've seen them. It exhibits a strong learning effect. People improve when retaking it more than on other IQ tests. Armstrong-Woodley claim that learning effect correlates with Flynn effect.
Intelligence seems to account for roughly 40% of the variance in the logarithms of mathematicians' research productivity, with the remainder accounted for by other innate abilities and environmental factors. This is consistent with most exceptionally intelligent mathematicians producing unremarkable math, and also (given the rarity of people with exceptional intelligence) consistent with some great mathematicians not being exceptionally intelligent. I'll write more about this later.
Nice to know there's still hope for the rest of us.
I've heard a version of this proposed as an explanation for the Flynn effect - industrialized urbanized nations with standardized schooling exposing people to more and more problems of the type the IQ test contains over time.
Some of my candidates (who, perhaps not coincidentally, also happen to be among my "favorite" old-time mathematicians, in the sense of stylistic identification):
All of these violate (what I think of as) the "math genius" stereotype in some way. None of these were considered child prodigies; in many cases they took up mathematics relatively late (Lie), had some competing interest (Cantor), or stood in contrast to a prodigy they knew (Hilbert, the prodigy being Minkowski).
Expanding the scope to physicists (and in the category of "widely held cultural beliefs that are probably wrong"), I will also nominate:
whom I suspect of possessing significantly less Tao-style ability, and being more akin to the above-listed mathematicians, than is commonly assumed.
Tao's abstract pattern recognition ability would seem to mark him as an outlier amongst mathematicians of similar accomplishment, whose relatively lower abstract pattern recognition abilities are counterbalanced by other abilities (some innate and others developed).