One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”
But Grothendieck must have known that 57 is not
prime, right? Absolutely not, said David Mumford
of Brown University. “He doesn’t think concretely.”
Consider by contrast the Indian mathematician
Ramanujan, who was intimately familiar with properties
of many numbers, some of them huge. That
way of thinking represents a world antipodal to that
of Grothendieck. “He really never worked on examples,”
Mumford observed. “I only understand things
through examples and then gradually make them
more abstract. I don’t think it helped Grothendieck
in the least to look at an example. He really got control
of the situation by thinking of it in absolutely
the most abstract possible way. It’s just very strange.
That’s the way his mind worked.” Norbert A’Campo
of the University of Basel once asked Grothendieck
about something related to the Platonic solids.
Grothendieck advised caution. The Platonic solids
are so beautiful and so exceptional, he said, that one
cannot assume such exceptional beauty will hold in
more general situations.
from "As If Summoned from the Void: The Life of Alexandre Grothendieck" by Allyn Jackson