dxu comments on Innate Mathematical Ability - Less Wrong
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I wasn't meaning to imply that you define success in terms of prizes (and, for that matter, neither do I). I agree that Langlands is a more important mathematician than Tao. But that's a hell of a bar to clear. (Also, speaking of age effects, I remark that if you define mathematical success in terms of what one has achieved to date and its demonstrated influence in mathematics generally, you're inevitably going to prefer older mathematicians -- Langlands is 78 to Tao's 40ish -- and that's going to affect what biases they have affecting their ideas about intelligence, native talent, etc.)
The quotation from Langlands that you give is not affirming the same thing as Tao is denying (though it's possible that Tao would in fact deny it if asked), in at least two ways.
For the avoidance of doubt, I won't be terribly surprised if it turns out that (say) 75% of world-class mathematicians think top 0.1% IQ is necessary to be a top 0.1% mathematician, but I'm not sure you've made much of a case yet. I'd be a little more surprised if it were 75% of world-class mathematicians who have put as much thought into the question as Tao has; I've no idea how much Langlands has actually thought about the question, but a throwaway aside in an essay about something else isn't necessarily the product of deep thought.
I'll briefly state my own opinions on all this, not that there's any reason why anyone should care. I'll use the term "R" for the particular thing that, e.g., Raven's matrices test. I think it's obvious that, all else being equal, more of any cognitive strength is better for mathematical success; more R is always going to be an advantage. So, of course, are other cognitive strengths, and other attributes such as love of mathematics, capacity for hard work, etc. They all doubtless correlate somewhat with R. So the question is something like: at a given overall level of rarity-of-useful-attributes, what things does mathematical success depend on most strongly? If you're looking at a population for which some particular attribute is extreme, then prima facie you would expect that attribute to drop off in importance by this criterion as it gets more extreme, because of how rarity increases. I think it's uncontroversial that to achieve any kind of mathematical success it's almost essential to have a pretty good R. Beyond a certain level -- let's say roughly corresponding to a measured IQ of 140 or so -- I think the importance of different cognitive strengths starts to depend a lot on what kind of mathematics you're doing. For instance, I would guess that combinatorialists tend to have higher IQ than differential geometers. That doesn't mean that combinatorialists are smarter than differential geometers, for two reasons. (1) I think equating R with smartness becomes less sensible at the highest levels; extreme R is a specialized talent and in terms of (e.g.) practical success, giving an immediate impression of great brainpower, etc., I strongly suspect that at a given overall level of rarity you're going to do better with very high R plus very high other things (e.g., Scott-style verbalish reasoning) than with extreme R optimized for acing IQ tests. (2) The particular bundle of talents needed for great success in differential geometry is probably about as rare, and about as clearly constitutive of "smartness" by any reasonable standard, as the particular bundle needed for great success in combinatorics. And beyond a certain level -- whose rarity maybe corresponds to a measured IQ of 150 or so -- at any given level of rarity I bet other attributes that aren't exactly cognitive strengths matter more than all those cognitive strengths do. So if you stratify prospective mathematicians by IQ (which of course is very similar to R), I would expect a picture like this. IQ <= 130: mathematical success very very strongly dependent on R. IQ <= 150: mathematical success correlated with R, but other cognitive strengths matter more in many fields of mathematics. IQ > 150: mathematical success mostly determined by other cognitive and not-so-cognitive strengths.
I conjecture that the above is broadly consistent with your evidence. I think it is also consistent (with a little interpretive licence -- e.g., his idea of what constitutes exceptional intelligence is probably skewed upwards relative to most people's) with what Tao has said. I am willing to be convinced that I'm wrong on either point.
Would you characterize this post as a reasonable description of what you're talking about in your discussion of "R"?
Yes, that's the guts of it.