Is there a rigorous definition of a line? In the sources I can find, line is considered a primitive concept (a one without definition). There is something, however, in such an intuitive definition that doesn't sit well with me.
Were there any attempts to rigorously define line?
And if there weren't, why? What makes line a concept which shouldn't be defined rigorously?

In the plane geometry that bears David Hilbert's name, there are five undefined terms:


*

*Line

*Point

*Lies on (as in a point lies on a plane) (this is often called "incidence")

*Between (as in one point is between two other points)

*Congruent (as in two line segments or two angles are congruent)


However, just because they are undefined does not mean that their behaviour is undefined. That is what all the axioms are there for.
From a logical standpoint, there is a reason that some terms must be undefined, and it's in two parts: First of all, definitions cannot be circular. Second, there are only finitely many words available to define any single concept. Together these two reasons makes it so that any chain of repeatedly asking "But what is the definition of that term?" is bound to eventually terminate in a standstill where you don't have any more words left to describe the concepts without resorting to circularity. Therefore, it is generally accepted that you just have to start somewhere, with names like "point" and "line" which you won't explain what is, just how they behave.
