[Alexandre] Grothendieck had a mathematical style all his own.... [I]t seemed completely different and new. But it is hard to articulate what the difference was....Strange, how minds work. I think there is a similar divergence of minds in law and in politics, but it's so nicely clear in that contrast of the mathematical minds, Ramanujan and Grothendieck.
Although Grothendieck approached problems from a very general point of view, he did so not for generality’s sake but because he was able to use generality in a very fruitful way....
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 toGrothendieck 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."
Lacking the subtle joy of living in a prime number year, I have the opportunity to see my life in segments, in this case, 3 segments of 19. (19 segments of 3 is not interesting.) The story of my life does, in fact, divide neatly into 19-year segments, not that I'm going to reveal why the first 2 19-year points look significant from the vantage point of the third. But will the fourth 19-year segment be different from the third? Will I have all 19 years? 19 beyond that? 19 beyond that? 19 beyond that? Surely, not 19 beyond that.