Outsiders
sometimes have an impression that mathematics consists of applying more and
more powerful tools to dig deeper and deeper into the unknown, like tunnelers
blasting through the rock with ever more powerful explosives. And that’s one
way to do it. But Grothendieck, who remade much of pure mathematics in his own
image in the 1960s and ’70s, had a different view: “The unknown thing to be known appeared to me as some
stretch of earth or hard marl, resisting penetration . . . the
sea advances insensibly in silence, nothing seems to happen, nothing moves, the
water is so far off you hardly hear it . . . yet it finally
surrounds the resistant substance.”
The unknown is a
stone in the sea, which obstructs our progress. We can try to pack dynamite in
the crevices of rock, detonate it, and repeat until the rock breaks apart, as
Buffon did with his complicated computations in calculus. Or you can take a
more contemplative approach, allowing your level of understanding gradually and
gently to rise, until after a time what appeared as an obstacle is overtopped
by the calm water, and is gone.
Mathematics as currently practiced is a delicate
interplay between monastic contemplation and blowing stuff up with dynamite.
