Should we think of lines as sets of points?

Question

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.

Answer 1

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.

Answer 2

Here's something I wrote about this viewpoint earlier:

A better answer to your misgivings (in the sense of being closer to the mainstream presentation) is probably simply to jump in with both feet and declare that a line is not really made of points.

Decide to think of a line as something that is in principle a different kind of thing from a bunch of points glued together. You can do this and still acknowledge that points exist and some of them are on the line while others are not.

It then turns out that all of the properties of a line segment (or a smooth curve in general) can be recovered from knowing which points lie on it and which don't. This doesn't necessarily mean that the points make up the line, but merely that the points tell us enough about the line.

It is technically convenient, then, to speak about the set of points on the line as a placeholder for the line itself, when we're formalizing our reasoning -- for the pragmatic reason that we have a well-developed common machinery and notation for speaking of sets of things, which means that we don't need to introduce a new formalism for an entirely different kind of things.

Some people are so comfortable with this representation that they happily declare that the line IS its set of points -- but nobody says you have to think of it that way. As long as you agree that the set of points determines the line, you can still communicate with people who prefer the other idea.

Why would one want to do that, you ask?

One reason is if it happens to be more comfortable philosophically for you to think that way. It's up to you whether it is or not, of course -- but, for example, the asker I wrote the above text for had gotten himself into conceptual trouble trying to imagine how points can have zero length and width, and yet combine to form a line of nonzero length.

Another reason is that you can avoid talking about sets at all for some purposes. Mathematicians use sets all the time, of course, but set theory does come at a foundational cost. For example, you can't have a formal theory that admits sets in a useful way without Gödel's incompleteness theorem applying, so the theory would necessarily be incomplete.

In particular, plain old Euclidean geometry can be formalized without having any concept of a set of points. That's what Euclid was doing, of course, though he missed a number of "intuitive" continuity properties. Hilbert tried to repair that, now explicitly considering "line" to be a primitive concept, but he failed to make it completely formal because formal logic was still being invented. (Later Tarski constructed an actual complete first-order theory of geometry, but he did it by omitting any concept of "line" at all, speaking instead of "collinear" as a relation between three points).