I have researched this question for days and can not locate a good answer. It could be a mathematical object that is defined by an axiom as Euclid or Hilbert. But if a curve is drawn between two points can it be should using only the rules of plane geometry that the curve is a "straight line"? If the curve is not a straight line then does it follow that the theorems that rely on such a construction will not necessarily be valid? BTW I am speaking of Euclidean geometry only.
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Euclid didn't define "line" by an axiom; in one translation, he defined it as "breadthless length," which is the kind of definition that only helps you if you already know what a line is. Isaac has it right when he suggests that what's important about a line is what it does, not what it is. An alternative approach is to found Euclidean geometry on analytic geometry. Define the real number line, then the Cartesian coordinate system, then define a line to be the set of solutions of $ax+by+c=0$ for fixed $a,b,c$ with $a$ and $b$ not both zero.
I am not entirely sure what you are asking, so this is likely a partial answer at best.
In many high-school-level texts in the U.S., line is one of several basic terms that are not formally defined. For example, in UCSMP Geometry, 3rd ed. (© 2009 Wright Group/McGraw-Hill), the first lesson in the text has:
Shortly thereafter, the Point-Line-Plane Postulate states:
Basically, a line is described by its properties but not formally defined.
When I was in Grammar School, our mathematics teacher defined a straight line as: "The shortest connection between two points which, when extended to infinity in either direction can never intersect with itself." This definition has always worked for me.