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:
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.
Be careful about using these wide inferential steps as an example. It is much easier to see certain consequences from a model than it is to prove consequences generated by a different model. It is a much better idea to practice deriving a (possibly different) set of consequences from a result you are comfortable with. This will give you a better idea of his intellect. Leading mathematicians often seem so much farther ahead than others because they are less constrained by the paths of other people.
I can see where Shelah is coming from, but if you’re quarter-way decent at compartmentalization and maybe work from two radically different examples (prime and composite, say, or if functions, odd and even functions, etc), it shouldn’t be a serious problem.
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:
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.
Be careful about using these wide inferential steps as an example. It is much easier to see certain consequences from a model than it is to prove consequences generated by a different model. It is a much better idea to practice deriving a (possibly different) set of consequences from a result you are comfortable with. This will give you a better idea of his intellect. Leading mathematicians often seem so much farther ahead than others because they are less constrained by the paths of other people.
I can see where Shelah is coming from, but if you’re quarter-way decent at compartmentalization and maybe work from two radically different examples (prime and composite, say, or if functions, odd and even functions, etc), it shouldn’t be a serious problem.