Should we think of lines as sets of points? I recently heard about a mathematician who denied that lines are sets of points, preferring to think of them as objects to which points may be incident.
What are the advantages of this point of view? These might be practical advantages or philosophical advantages. I have a feeling that it may be less practical than the common idea of lines as sets of points. For example, if we want to consider a bijection from a line to the real numbers, we now have to consider the set of all points incident to the line.

The "advantage" of thinking of points and lines in geometry as primitive objects related by incidence (I classify this as synthetic geometry) is that it is very general and easy to get started. The board is relatively free of clutter compared to starting the other way with "a plane is a set of points, and there are special subsets called lines with such and such properties..." and perhaps with something like "...and points have coordinates using $\mathbb R$..." (this is what I would call coordinate or analytic geometry.) You have to be familiar with the language of set theory to model everything with sets. 
But for people who are comfortable with sets, sets are a very natural model for points, lines, and incidence.
One more thing: if you only care about Desarguesian geometry, you are free to adopt either viewpoint. This is the case for Euclidean metric  geometry, and yes, it is usually helpful to associate sets and measurements with geometric objects when first leaning geometry. In fact, plain synthetic geometry leads to more exotic geometries than those encodeable by coordinates over fields or even division rings. That is why I mentioned that it is more general than coordinate geometry. 
