I confess I am struggling to make sense of this. If by a "geometric point" you mean that, for example, every point in \(E^2\) can be regarded as the vertex of some arbitrary n-gon, then each vertex has that very same "structure" i.e the vertex of some n-gon, with edges and angles (presumably) easily found. On the other hand, if you simply mean to imply that a "geometric point" has "no structure" whereas a point \(p = (x,y) \in R^2\) does have some "structure" then I would ask what that "structure" might be. I apologize for being argumentative, but I really and truly don't get it
Hmm. In \( \mathbb E^2 \) you can construct a set of non-intersecting lines, and these are then all in "general position". Then the endpoints of each line are just "places", each could be labelled with some unique symbol, as could each line. But there is no "geometry" as such [of the analytic variety], since you have lines with arbitrary size, arranged at arbitrary angles and no coordinates. If you construct say, an equilateral triangle you can label each vertex (resp. edge) with some unique symbol and then you have planar symmetry, but again geometry isn't really in there. (?) [Analytic] Geometry requires distances being defined, and that means each point must have coordinates, so then you have to really be in \( \mathbb R^2 \)
I'm going with the following simple version: if you use a straight edge which isn't ruled and a compass, you can construct all kinds of geometric figures, but they are all defined by intersections of lines and these intersections can be given arbitrary labels. In other words if you do this: Please Register or Log in to view the hidden image! . . . then you can add a coordinate system and determine "real" lengths. But without coordinates you already have that AB is the radius of both circles, and A,B are two of the "geometric points" rpenner refers to. As to what "structure" means in this context, I'll leave that to someone else.
