Two ways to crack a walnut, per Grothendieck (2025)

Summary

The mathematician Alexander Grothendieck was “considered by many to be the greatest mathematician of the twentieth century”. Somewhere in his 1000+-page autobiographical work Récoltes et Semailles (“Harvests and Sowings”), he describes two styles in mathematics: Take for example the task of proving a theorem that remains hypothetical (to which, for some, mathematical work seems to be reduced). I see two extreme approaches to doing this. One is that of the hammer and chisel, when the problem posed is seen as a large nut, hard and smooth, whose interior must be reached, the nourishing flesh protected by the shell. The principle is simple: you put the cutting edge of the chisel against the shell, and hit it hard. If necessary, you repeat the process in several different places, until the shell cracks—and you are satisfied. He goes on to describe this a bit, but he favoured a second approach: I can illustrate the second approach with the same image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months—when the time is ripe, a touch of the hand is enough, and the shell opens like a perfectly ripened avocado! He has even more imagery for this second approach: A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration. One can go at it with pickaxes or crowbars or even jackhammers: this is the first approach, that of the “chisel” (with or without a hammer). The other is the sea. The sea advances insensibly and 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. (Translations merged from Tong Zhou Part III here, Colin McLarty here, and cag51 here.) This was Grot...