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.
The point I was trying to make is that it may not be necessary to have “a large stack of examples”. It might instead be much more useful to have a couple of “protoypal concrete examples...a root example”. Kontsevich seems to have similar thought patterns.
I don’t know about Grothendieck. But, Kontsevich’s statement is telling:
Halmos (quoting Hilbert) captures this very well:
I already mentioned what Halmos’ stance was. What I’m more interested in is how is it possible to work without examples.
The point I was trying to make is that it may not be necessary to have “a large stack of examples”. It might instead be much more useful to have a couple of “protoypal concrete examples...a root example”. Kontsevich seems to have similar thought patterns.