Two ways to crack a walnut, per Grothendieck (2025)
shreevatsa.net
shreevatsa.net
Dyson calls the mathematicians solving such problems frogs, respectively birds. https://www.ams.org/notices/200902/rtx090200212p.pdf:
“Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the far horizon. They delight in con- cepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects, and they solve problems one at a time.”
"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."
(From _Récoltes et Semailles_)
It's only a shame that most aspiring mathematicians with the same disposition as Grothendieck, never do reach the same level of accomplishment and acclaim as Grothendieck, before concluding that they are, in their final opinion, like clumsy, oafish oxen compared to their colleagues.
I know someone who believes he has solved all the remaining major math and physics challenges and submitted them for the various awards.
He is alone, too. His schizophrenia has isolated him pretty thoroughly.
The idea that Grothendieck both massively succeeded and failed in some sort of countercultural/neurodivergent knife edge is I think the ambiguous but correct morality tale.
It was a let-down: they were easier to open, but certainly not by hand, I still had to use a nutcracker. They smelled bad, too. I did not try eating them.