If anyone's curious about the actual math Grothendieck was doing but is (like me) a layman when it comes to mathematics, there's a useful summary of one of his major areas of work in parts II and III of this blog post [0].
The essence, I believe, is
>Bottom line: To study the Weil conjectures, you have to think very hard about the subtle properties of curves, surfaces and higher-dimensional objects. When you do this, you find yourself mentally “moving around” the curve, trying to hop from point to point. And — in a sense that I cannot hope to make precise here — you sometimes find that your mental exploration is hampered by the fact that there somehow aren’t enough points to hop to.
>So life would be easier if these curves (and other objects) had more points. A normal person might say “Well, life’s not always easy. We’ll just have to get by somehow with the points we’ve got”. But Grothendieck lived by the conviction that everything is easy if you look at it right — which means there have got to be enough points. And if we think there aren’t, it must be because we haven’t yet figured out what a point is.
I don't think I'm closer to understanding what it is he was working on after reading it, at least not in a meaningful sense, but it is fascinating to me to read about.
The metaphor cited below about the rising sea softening up the nut for you is a very apt way of characterizing Grothendieck's way of thinking.
In the particular example that your quote tries to explain, he tackled the following problem: In the geometry of the complex numbers, there is a good way of thinking about 'small' neighborhoods of a point, for instance all points that are fixed very small distance ('epsilon') away.
Unfortunately, this doesn't make sense algebraically: there is no polynomial equation that codifies being a 'small' distance away. What Grothendieck did was to essentially define away the problem by defining algebraic 'neighborhoods' (étale neighborhoods) that recover the complex analytic notion. These are no longer actual neighborhoods (they're not subsets) but rather spaces mapping to your original one.
The key insight here is that neighborhoods don't have to be a subspace of the space you start with. Once you allow your mind this freedom, then everything works out (with quite a bit more work of course!)
I was just about to share the very same article above after reading your comment just to check that it's the same.
My favorite part of the article, to which I as a mathematician relate a lot on my a-ha moments:
> If there was a nut to be opened, Grothendieck suggested, Serre would find just the right spot to insert a chisel, he’d strike hard and deftly, and if necessary, he’d repeat the process until the nut cracked open. Grothendieck, by contrast, preferred to immerse the nut in the ocean and let time pass. “The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough.”
I don't know Grothendieck's work, but this sounds like "pointless topology" for which I know a pretty good example (I also have Johnstone's book but it's terribly difficult for me to get into).
Consider the "set" S of all surjections from the natural numbers to the real numbers. It should be immediately clear that no such surjections exist, thus we have a "set" of no points. That said, suppose we'd like to talk about the properties of this set nonetheless. Are there any?
Instead of working concretely, we can work logically. Instead of discussing individual points, I'll name properties these points might have should they exist. For instance, I could talk about all of the subsets of S such that (s in S) has that s(n) = r for some values n and r. There are NxR such properties, one for each choice of n and r.
These properties can also be combined. I can talk about (n1, r1) AND (n2, r2). They in fact form open subsets of S and thus a topology of this "pointless" set S.
So in this way, I actually can discuss quite a lot about S despite it having no concrete points. I can even construct another set S' which does have all of the points needed to support the "properties" I was discussing earlier. S and S' are related in that much of what we can say about S' we can "port" over to S, so in some sense the points of S' are "phantom points" of S.
And this sort of thing is what I think Grothendieck began doing.
I have fair amount of mathematical training, including some graduate degrees, and Grothendieck's mathematical contributions (other than his very early stuff) are still an enigma to me. To truly grok his major accomplishments you have to go waaay down the abstract algebra rabbit hole.
This seems to be an unfair reading of that sentence. A fairer interpretation might be closer to "right tool for the right job". Certain topics or materials can be more easily understood when viewed with the right framework. In physics, for instance, calculus yields results that mere algebra and trigonometry cannot easily provide. It's not that calculus is easy, but that with it (the framework, point of view), the physics becomes easy.
EDIT: After reading more, I stand by this. And Grothendieck seemed to specialize in creating those frameworks.
Aside from frameworks looking at it right might have to come from years of study which may necessarily not be easy but once you find something easy you are looking at it right.
> He ignores the silent and flawless consensus that is part of the air we breathe – the consensus of all the people who are, or are reputed to be, reasonable.
Very well put, especially considering this is from a personal journal.
The essence, I believe, is
>Bottom line: To study the Weil conjectures, you have to think very hard about the subtle properties of curves, surfaces and higher-dimensional objects. When you do this, you find yourself mentally “moving around” the curve, trying to hop from point to point. And — in a sense that I cannot hope to make precise here — you sometimes find that your mental exploration is hampered by the fact that there somehow aren’t enough points to hop to.
>So life would be easier if these curves (and other objects) had more points. A normal person might say “Well, life’s not always easy. We’ll just have to get by somehow with the points we’ve got”. But Grothendieck lived by the conviction that everything is easy if you look at it right — which means there have got to be enough points. And if we think there aren’t, it must be because we haven’t yet figured out what a point is.
I don't think I'm closer to understanding what it is he was working on after reading it, at least not in a meaningful sense, but it is fascinating to me to read about.
[0] http://www.thebigquestions.com/2014/11/17/the-generalist/