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.
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.