“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.”
Every time I refactor a bunch and then the new feature just falls out for free, I solute Grothendieck. It's baby-mode, but it's still the rising sea method.
Another quote of Grothendieck's that has always stuck with me:
"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.
"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 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.
Grothendieck ended up totally crazy too, unfortunately. Some people might be truthers that his final writings would be made sense of someday, but I don't think that is a responsible hope to carry.
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.
The most gifted scientists are of one of two flavors, seemingly: Completely lucid and exuberant well into their elderly years, or strange and secluded even in their better years.
The Grothendieck method can be applied to implementing a new feature in a software system. You just refactor the existing code until the implementation of the new feature becomes trivial.
When reading code you don't really understand, that approach looks a lot like magic. I have a few code bases I try to follow, one is a large Java application, and some of the features added seemingly come out of nowhere. The commit message will just say "Add support for XYZ" and then a few lines added to seemingly random files to setup of condition and then the feature manifests itself.
I wonder how the actual way of opening a walnut fits into this metaphor: apply pressure to the gap line of the two halves. You can even do it with your bare hands a few times. Do not use a chisel, the nut will slip easily, duh.
On the topic literally: I found this holiday that a flat head screwdriver works best. There’s always a small gap and just prying the nut open is far better than crunching it and making a mess.
Though I was confident this was just a metaphor, I did try cracking a walnut using this method. I submerged three walnuts in a jar with some tap water, and let them sit for three years before trying to open them.
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.
FTA: “Some problems benefit from zooming in, others from zooming out. Grothendieck was the messiah of zooming out”
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.”
Every time I refactor a bunch and then the new feature just falls out for free, I solute Grothendieck. It's baby-mode, but it's still the rising sea method.
Another quote of Grothendieck's that has always stuck with me:
"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.
"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 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.
Grothendieck ended up totally crazy too, unfortunately. Some people might be truthers that his final writings would be made sense of someday, but I don't think that is a responsible hope to carry.
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.
The most gifted scientists are of one of two flavors, seemingly: Completely lucid and exuberant well into their elderly years, or strange and secluded even in their better years.
The Grothendieck method can be applied to implementing a new feature in a software system. You just refactor the existing code until the implementation of the new feature becomes trivial.
When reading code you don't really understand, that approach looks a lot like magic. I have a few code bases I try to follow, one is a large Java application, and some of the features added seemingly come out of nowhere. The commit message will just say "Add support for XYZ" and then a few lines added to seemingly random files to setup of condition and then the feature manifests itself.
I wonder how the actual way of opening a walnut fits into this metaphor: apply pressure to the gap line of the two halves. You can even do it with your bare hands a few times. Do not use a chisel, the nut will slip easily, duh.
I keep telling myself I'm using the sea approach but my git history says hammer and chisel
Oh! I thought it's going to be about some interesting topology of nut shell...
Likewise, I'd assumed sphere eversion: https://en.wikipedia.org/wiki/Sphere_eversion
That's ... concerning.
On the topic literally: I found this holiday that a flat head screwdriver works best. There’s always a small gap and just prying the nut open is far better than crunching it and making a mess.
I get the point of the story but does the soaking method for opening walnuts actually work? Seems to me that you'd likely end up with a rotten nut.
Though I was confident this was just a metaphor, I did try cracking a walnut using this method. I submerged three walnuts in a jar with some tap water, and let them sit for three years before trying to open them.
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.
It is a mixed metaphor, he refers to the problem as a "nut" but says after soaking the nut should open "like a ripe avocado"