Contrary data point here:

That's very interesting to me – thanks for sharing.

. When he says "exceptional" here, I think he means it in the ordinary sense of the word -- the sense relevant to most of the readers he's addressing -- which would include not only himself but also almost all of his UCLA colleagues (for example).

Thanks for pointing out a possible alternative explanation. Can you elaborate? I think that I might understand what you're saying, but I'm not sure. Are you saying that UCLA math professors would be considered to be exceptional mathematicians but not exceptionally intelligent? It's not clear to me that this is the case – you seem to be breaking symmetry by interpreting his two uses of 'exceptional' in different ways.

UCLA math professors are as a group more intelligent than UCLA math grad students, who are in turn as a group more intelligent than UCLA math majors. His remarks in the article that I linked suggests that he adheres to the threshold theory – that after a certain point intelligence doesn't yield incremental returns. I think that this is wrong whatever reference class one is using.

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.

It's not entirely clear to me how somebody as mathematically talented as Tao could miss the basic Bayesian probabilistic argument that Scott Alexander gave, which shows that Tao's own existence is very strong evidence against his claim. But two hypotheses come to mind.

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.

Very interesting, thanks!

I'll have more to say about the role of verbal reasoning ability in math later on

When you do, I hope you'll mention Paul Halmos, one of my favorite mathematicians (and the author, among many other things, of Naive Set Theory, which is on the MIRI reading list), who famously began his autobiography with the sentence "I like words more than numbers, and I always did."

People who are able to pick the correct choice at all can usually do so within 2 minutes – the questions have the character "either you see it or you don't."

Contrary data point here: I eventually figured out the "correct" answer (in the sense of the answer that everyone else came up with), but it took me something like 15-20 minutes (including interruptions by various distractions, such as reading subsequent paragraphs -- which I'm glad I did, because it allowed me to discover that the test was untimed, which is what gave me the confidence to try to figure it out!).

A reasonable amount of intelligence is certainly a necessary (though not sufficient) condition to be a reasonable mathematician. But an exceptional amount of intelligence has almost no bearing on whether one is an exceptional mathematician.

It's not entirely clear to me how somebody as mathematically talented as Tao could miss the basic Bayesian probabilistic argument that Scott Alexander gave, which shows that Tao's own existence is very strong evidence against his claim.

I think this is uncharitable to Tao. When he says "exceptional" here, I think he means it in the ordinary sense of the word -- the sense relevant to most of the readers he's addressing -- which would include not only himself but also almost all of his UCLA colleagues (for example).

When you say "The heliocentric view had only a single advantage against the geocentric one: it could describe the motion of the planets by a much simper formula," you falsely do it a service, because you overstate the practical differences. Galileo was no Kepler - he didn't have improved observational accuracy. The Copernican model and the Tychonic model (adopted by the Catholic church only a decade and change before Galileo published the Dialogue) make basically the same predictions for planetary movement.

This means the really interesting lie is "The geocentric view had a very simple explanation, dating back to Aristotle: it is the nature of all objects that they strive towards the center of the world, and the center of the spherical Earth is the center of the world. The heliocentric theory couldn't counter this argument." Because Galileo's answer is what this was really about! If Tycho Brahe was right, Earth was made of a lazy type of matter, which desired to be at rest and fall down, while the heavens followed an entirely different set of rules that demanded perpetual motion. But if Galileo was right, an object in motion would stay in motion unless acted upon by an outside force, whether on earth or in the heavens.

After 1610 or so, if one had a decent understanding of astronomy, one accepted that Ptolemy was wrong and that heliocentrism was now not about overthrowing the Greeks' astronomy, but their mechanics. Who was right? Was it Aristotle and Brahe? Or was it Descartes and Galileo? Of course, the discussion would have been a lot more interesting if the church hadn't started banning the books of one side after they declared heliocentrism heretical, but oh well.

I've noticed in courses I taught that grades tend to reward conscientious students who can "play the game" and do formal manipulations even if they don't really understand what's going on.

Calculus 2 is where I hit the limits of my conceptual abilities. I am very bad at "playing the game" in this way, so I haven't moved beyond that yet.

I think it's wrong to put too much emphasis on a contrast between "playing the game" and "understanding the material", though. My feeling is that if I became better at playing games, paying attention to detail, being more conscientious about my work, then I would also improve my conceptual understanding after a while.

I'm extremely grateful for this post, and look forward to the rest of the sequence.

For me this is also of great personal relevance -- I too am among the "twice exceptional" (*), and am chagrined that this concept, as the Wikipedia article says, "has only recently entered educators' lexicon". You won't be surprised to know that (as I think we've even discussed before privately) Grothendieck's description of himself -- and his mathematical style, insofar as I understand it -- is also something that I identify with very strongly.

(*) illustrative anecdote: in 9th grade, I received a "D" in geometry during the same term that I won a state competition in that subject.

I don't know Latin so I'm guessing "lecture" is Latin for "lost purpose"? That's great, thanks for educating me.

Taken.

I personally have no problem with that -- but komponisto wants to make more detailed distinctions, and was originally (i.e., at the other end of the link in the great-grandparent of this comment) responding to someone else who wanted to count courses currently in progress as well as ones already completed.

I'm sure both of them have reasons (indeed, it's not hard to guess some) and I bet they're both aware that it's usual simply to ask for highest qualification actually attained.

I shall repeat my request for a second question under "Degree" that asks about one's highest degree attempted or in progress.