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 Grothendieck’s approach, which worked very well for him:
Deligne describes a characteristic Grothendieck proof as a long series of trivial steps where “nothing seems to happen, and yet at the end a highly non-trivial theorem is there”.
Grothendieck also wrote in a letter:
The question you raise, “how can such a formulation lead to computations?” doesn’t bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand — and it always turned out that understanding was all that mattered.
There are some remarks one can make on top of this.
Although Grothendieck had very great success with the second approach, he himself said that Serre, who (in his view) generally uses the hammer and chisel, was the “incarnation of elegance” — Serre concisely cuts to an answer.
There are limitations to the second approach—there are problems for which it absolutely will not work. Serre himself raised this possibility, as Steven Landsburg points out here after:
Now, not all problems are like that. Some problems benefit from zooming in, others from zooming out. Grothendieck was the messiah of zooming out — zooming out farther and faster and grander than anyone else would have dared to, always and everywhere. And by luck or by shrewdness, the problems he threw himself into were, time after time, precisely the problems where the zooming-out strategy, pursued apparently past the point of ridiculousness, led to spectacular, unprecedented, indescribable success. As a result, mathematicians today routinely zoom out farther and faster than anyone prior to Grothendieck would have deemed sensible. And sometimes it pays off big.
Finally, Grothendieck’s approach only works when one is really good. If we call the first (not Grothendieck’s) approach “problem-solving” (The Two Cultures of Mathematics), then:
An excellent problem-solver might reach certain limits as a mathematician, while a bad problem-solver can still be an okay mathematician. On the other hand, a good Grothendieck can be a truly great mathematician, while a bad Grothendieck is really terrible! — Greg Kuperberg’s friend, paraphrased
So, as much as I like this quote and as awesome as Grothendieck’s approach sounds, it’s remembering that sometimes we need the opposite reminder too: sometimes finishing things quickly can be better than “analysis paralysis” for a long time.
(And it can also be a terrible approach at work, as I found out.)