Alexander Grothendieck, a genius of mathematics (2015)

so I can measure my arrogance in nanoGrothendiecks, cute.

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.”

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!)

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.