Another example is that Saharon Shelah, perhaps the most accomplished living set theorist/logician, is known to disdain examples. Here's one expression of his view:

I have always felt that examples usually just confuse you (though not always), having always specific properties that are traps as they do not hold in general.

My opinion is that Grothendieck, Kontsevich and Shelah are not fooling themselves, but their advice is wrong for most people who are not them. Personally, I'm annoyed at myself at not remembering more often to approach unfamiliar claims through small and simple examples. Whenever I do, I end up understanding the general claim more clearly, nonwithstanding the warning about the specific properties of examples. When I do not, I often end up feeling as if I'm in a fog, grasping perhaps the literal meaning of the claim but not being able to see its significance or what it implies.

Perhaps those exceptional people who dislike examples (and I don't think that this view is typical among even the most accomplished mathematicians) get that clarity of understanding from the claim itself, and don't need to unfog their brain through looking at examples. I could believe that in the case of Shelah, anyhow. I took his advanced course in set theory once, a long time ago. It was the closest I've ever come in my life to feeling that I've encountered not just someone much smarter than me, but a truly superior intellect from a whole different level. His atomic inferential step was unbelievably wide - that is, he (genuinely and humbly, without any attempt at showing-off) saw as an immediate consequence something it took a hard effort of several minutes for others to work through, again and again. It was incredible to watch, and I've never seen anything like that with any other mathematician.

I feel reminded of the discussion whether you can actually see something you imagine. Turns out, some people can see a vivid image in front of their eyes and some are simply incapable of this. Maybe this is a similar case: Some people work well with examples, some are better with general abstract concepts.

P.S.: I feel this is a good time to say this: In my limited experience teaching and learning I found that people are massively inhomogenous with regard to how they grasp concepts and there is no telling which way is obviously better. I assume that proper didactics would address different ways of learning and working.

I suspect Groethendieck was an alien.

For myself, I agree with Scott, I run into trouble without a ton of examples. You eventually get a feel for when the list of examples is "good enough" for the general case.

I don't think it is strictly true that Grothendieck didn't rely on examples. Here is a quote from a letter from Luc Illusie, who was a grad student under Grothendieck. Quote:

“In his filing cabinets, located behind his desk, Grothendieck kept many handwritten notes, where he had studied specific examples: he sometimes told me that he was weak on surfaces, but as everybody knows, he was not so weak in local algebra, and he knew enough of curves, abelian varieties and algebraic groups to be able to test his ideas. Also, his familiarity (and constant interest) in analysis and topology was a strong asset. All these examples appeared when you discussed with him."

Slightly related ... there is a recent philosophy of mathematics book (albeit a work of continental philosophy, but it is still interesting). I say slightly related, because firstly it has an entire chapter on Grothendieck and creativity, and secondly the entire book is about the interplay between examples and theory. It goes into the way mathematics moves up and down a ladder of abstraction (from particulars to universals, local to global, specialization to generalization). It focuses on the last 100 years of mathematics, concentrating on algebraic geometry, category theory, model theory, and others. The author splits up the abstractions into:

• Eidal mathematics, or "transfusions of form." This is mathematics that moves up the ladder from the particular to the universal. In this part of the book it focuses on the work of Serres, Langlands, Lawvere, Shelah.
• Quiddital mathematics, or "transfusions of reality." This is abstractions made concrete moving down the ladder. Here the book focuses on Atiyah, Lax, Connes, Kontsevich.
• Archeal mathematics, or "decantations of the universal." This is mathematics that studies the invariants while one moves up and down the ladder. Here it focuses on Freyd, Simpson, Zilber, Gromov.

Now you could say this philosopher is just dressing up stuff already known to mathematicians (Both Kevin Houston's How to think like a mathematician, and Mason/Burton's Thinking Mathematically talk about generalization and specialization). But it is still interesting, as I think it may be the only philosophy of mathematics book out there that does an indepth treatment of 20th and 21st Century mathematics that goes beyond Russell and Frege.

Of my two friends who I talk the most math with, I explain to one of them with examples and the other with general statements. I don't know why it helps, I just know from experience that I have to give examples to one and general statements to the other.

It reminds me of the witch powers from Alicorn's Twilight fanfic. Some witches have power over the same thing, experienced through different senses.

Such witches were especially powerful working in pairs. I wonder if there's anything special about collaborations between example thinkers and general thinkers? I'm an example thinker, and with my general thinker friend, a common pattern is that he'll conjecture, and I'll search for counterexamples.

In both of quotes the non-example users are presented as unusual, right? So that doesn't really seem to contradict Aaronson's advice.

Grothendieck's mind was indeed extremely strange. The levels of abstraction upon abstraction he achieved in algebraic geometry boggles the mind.

But I don't think you can really make meaningful comparisons between thought processes based on self-reporting. One complication is that different fields of mathematics work differently in this regard. In things like statistics, analysis, and geometry, you rely heavily on examples. In things like algebra, examples can indeed be cumbersome and hindering, because the point of algebra is to simplify things to symbol manipulation. Of course, it might also be the case that people with more abstract-type thinking are naturally drawn to algebra.

It would be useful to look at the 'information content' of storing examples vs. storing symbolic representations, and see how that compares across different mathematical subjects.

I don't know about Grothendieck. But, Kontsevich's statement is telling:

For myself sometimes I work on one or two examples

Halmos (quoting Hilbert) captures this very well:

What mathematics is really all about is solving concrete problems. Hilbert once said (but I can't remember where) that the best way to understand a theory is to find, and then to study, a prototypal concrete example of that theory, a root example that illustrates everything that can happen. The biggest fault of many students, even good ones, is that although they might be able to spout correct statements of theorems, and remember correct proofs, they cannot give examples, construct counterexamples, and solve special problems.

I shall discuss many concepts, later in the book, of a similar nature to these. They are puzzling if you try to understand them concretely, but they lose their mystery when you relax, stop worrying about what they are, and use the abstract method.

Timothy Gowers in Mathematics: A Very Short Introduction, p. 34