Alexander Grothendieck, one of the most influential mathematicians of the 20th century, on cognitive "facility" and giftedness:
Since then I’ve had the chance in the world of mathematics that bid me welcome,
to meet quite a number of people, both among my “elders” and among young people
in my general age group who were more brilliant, much more ‘gifted’ than I was.
I admired the facility with which they picked up, as if at play, new ideas,
juggling them as if familiar with them from the cradle–while for myself I felt
clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox
faced with an amorphous mountain of things I had to learn (so I was assured)
things I felt incapable of understanding the essentials or following through to
the end. Indeed, there was little about me that identified the kind of bright
student who wins at prestigious competitions or assimilates almost by sleight
of hand, the most forbidding subjects.
In fact, most of these comrades who I gauged to be more brilliant than I have
gone on to become distinguished mathematicians. Still from the perspective or
thirty or thirty five years, I can state that their imprint upon the
mathematics of our time has not been very profound. They’ve done all things,
often beautiful things in a context that was already set out before them, which
they had no inclination to disturb. Without being aware of it, they’ve remained
prisoners of those invisible and despotic circles which delimit the universe of
a certain milieu in a given era. To have broken these bounds they would have to
rediscover in themselves that capability which was their birthright, as it was
mine: The capacity to be alone.
Since then I’ve had the chance in the world of mathematics that bid me welcome,
to meet quite a number of people, both among my “elders” and among young people
in my general age group who were more brilliant, much more ‘gifted’ than I was.
I admired the facility with which they picked up, as if at play, new ideas,
juggling them as if familiar with them from the cradle–while for myself I felt
clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox
faced with an amorphous mountain of things I had to learn (so I was assured)
things I felt incapable of understanding the essentials or following through to
the end. Indeed, there was little about me that identified the kind of bright
student who wins at prestigious competitions or assimilates almost by sleight
of hand, the most forbidding subjects.
In fact, most of these comrades who I gauged to be more brilliant than I have
gone on to become distinguished mathematicians. Still from the perspective or
thirty or thirty five years, I can state that their imprint upon the
mathematics of our time has not been very profound. They’ve done all things,
often beautiful things in a context that was already set out before them, which
they had no inclination to disturb. Without being aware of it, they’ve remained
prisoners of those invisible and despotic circles which delimit the universe of
a certain milieu in a given era. To have broken these bounds they would have to
rediscover in themselves that capability which was their birthright, as it was
mine: The capacity to be alone.
They are. Or at least, this guy is. Quote from Dieudonne, his doctoral advisor, describing how Grothendieck got his PhD:
>A general theory of duality for locally convex spaces had to be worked out: Schwartz and I had started its study for Fréchet spaces and their direct limits, but we had met a series of problems we could not solve. We therefore proposed them to Grothendieck, and the result turned out to exceed our most sanguine expectations. In less than a year, he had solved all our problems by very ingenious new constructions; then, with the techniques he had developed, he started to work on many other questions in functional analysis.
