[lmarinho]

Good point, my statement was poorly formulated. While I agree that analytic geometry and other developments in the field could be more fundamental, Hilbert's formalization can still be useful to express more precisely what one means by "points" and "lines", being, I think, applicable to the current discussion.

[Sniffnoy]

On the contrary, it's an axiomatization, and thus only describes the relation between them. If by "what one means by them" you mean a definition, then you'll need a construction for that. Of course, for the most part, the relation between them is what we mean by them; but Hilbert's formalism doesn't seem to be a good way to address such statements as the one that started this, "a point is a line segment of zero length".