I’d never encountered this exquisitely characterizing passage from Grothendieck’s memoir before. I think even non-mathematicians will find it of interest.
In those critical years I learned how to be alone.[But even]this formulation doesn’t really capture my meaning. I didn’t, in any literal sense, learn to be alone, for the simple reason that this knowledge had never been unlearned during my childhood. It is a basic capacity in all of us from the day of our birth. However these three years of work in isolation[1945-1948],when I was thrown onto my own resources, following guidelines which I myself had spontaneously invented, instilled in me a strong degree of confidence, unassuming yet enduring in my ability to do mathematics, which owes nothing to any consensus or to the fashions which pass as law..By this I mean to say: to reach out in my own way to the things I wished to learn, rather than relying on the notions of the consensus, overt or tacit, coming from a more or less extended clan of which I found myself a member. or which for any other reason laid claim to be taken as an authority. This silent consensus had informed me both at the lycee and at the university, that one shouldn’t bother worrying about what was really meant when using a term like” volume” which was “obviously self-evident”, “generally known,” ”in problematic” etc…it is in this gesture of ”going beyond to be in oneself rather than the pawn of a consensus, the refusal to stay within a rigid circle that others have drawn around one-it is in this solitary act that one finds true creativity. All others things follow as a matter of course.
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.
I’ll add just one remark: “The capacity to be alone” is a phrase made famous by the psychoanalyst D.W. Winnicott, who understood the development of this capacity to be a crucial phase in the maturation of the child. Winnicott’s sense of the term is quite specific: “the basis of the capacity to be alone is a paradox; it is the experience of being alone while someone else is present.” I don’t know whether Grothendieck was quoting Winnicott here (is it known whether he was analyzed, or familiar with the psychoanalytic literature at all?) but his sense of the phrase is much the same. The challenge is not to do mathematics in isolation, but to preserve a necessary circle of isolation and autonomy around oneself even while part of a mathematical community.
I should say that this is totally foreign to my own mode of mathematical work, which involves near-constant communication with collaborators and other colleagues and a close attention to the “notions of the consensus,” which I find are usually quite useful.
Also, Grothendieck’s distinction between himself and the less profound mathematicians who were quick studies and winners of competitions should give John Tierney something to think about.