This approach to problems is impressive, but it’s impressive in the same way that a man doing complex calculations without a computer is. It seems tedious beyond reason for someone able to abstract better.
Huh?? I think mathematicians and physicists that are worth their salt think of examples and keep them in mind. It is common advice in math to think of specific examples. When I read a different quote of his about keeping examples in mind when mathematicians told him theorems and checking them against the object in his head I thought “duh—the real thing going on is that either you were better at thinking of an example, or the conjecture was brand spanking new, or the person talking about it is unusually bad”
Edit: The real quote was:
I had a scheme, which I still use today when somebody is explaining something that I’m trying to understand: I keep making up examples. For instance, the mathematicians would come in with a terrific theorem, and they’re all excited. As they’re telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball) — disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn’t true for my hairy green ball thing, so I say, “False!”
If it’s true, they get all excited, and I let them go on for a while. Then I point out my counterexample.
“Oh. We forgot to tell you that it’s Class 2 Hausdorff homomorphic.”
“Well, then,” I say, “It’s trivial! It’s trivial!” By that time I know which way it goes, even though I don’t know what Hausdorff homomorphic means.
So, not because of being fresh-of-the-shelf or incompetence, rather on not pointing out a technical condition.
Huh?? I think mathematicians and physicists that are worth their salt think of examples and keep them in mind. It is common advice in math to think of specific examples.
When I read a different quote of his about keeping examples in mind when mathematicians told him theorems and checking them against the object in his head I thought “duh—the real thing going on is that either you were better at thinking of an example, or the conjecture was brand spanking new, or the person talking about it is unusually bad”Edit: The real quote was:
So, not because of being fresh-of-the-shelf or incompetence, rather on not pointing out a technical condition.