The Sea and the Stone
This mental model contrasts two approaches to overcoming obstacles: the forceful, explosive method (like dynamite) versus the gentle, contemplative method (like rising water). It suggests that sometimes allowing understanding to gradually rise can overcome resistant problems without forceful effort.
THE IDEA
The Sea and the Stone
Barbier’s proof reminds me of what the algebraic geometer Pierre Deligne wrote of his teacher, Alexander Grothendieck: "Nothing seems to happen, and yet at the end a highly nontrivial theorem is there."
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 potent 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.
When to use it
Tackling a complex mathematical proof that resists direct computation
Navigating a strategic business challenge where initial aggressive moves have failed
Resolving a personal conflict by slowly building mutual understanding rather than confronting directly
Learning a new skill that requires deep comprehension over rote memorization
FROM THEORY TO EVERYDAY LIFE
In practice
Buffon's Forceful Calculus
Georges-Louis Leclerc, Comte de Buffon, used complicated computations in calculus to attack problems directly, akin to packing dynamite into crevices and detonating it repeatedly until the rock breaks apart.
Grothendieck's Contemplative Revolution
Alexander Grothendieck reshaped pure mathematics by allowing his understanding to rise gradually, like the sea surrounding a stone, leading to profound insights without brute force.
A MOMENT TO REFLECT
Put it to work.
Think of a problem you've been trying to solve with a forceful approach. How might a more contemplative, patient strategy change the outcome?
When encountering a stubborn obstacle, how can you tell whether to apply more force or to step back and let your understanding rise?
What practices can help you cultivate the patience needed for the 'sea' approach in your own work or life?
CONTINUE THE CONVERSATION
Source texts
- How Not to Be Wrong The Power of Mathematical Thinking (Ellenberg, Jordan)