I'm preparing geometry classes and I thought it is a good time to answer a question I had when I started to study geometry: why, in Euclidean axiomatic geometry, is the notion of a straight line primitive? Wouldn't it be more natural to take line segments as a primitive notion? When drawing, we draw line segments (approximately), and, of course, it is much more comfortable to imagine a (finite) line segment than an entire (infinite) straight line. Is there any philosophical reason why lines were taken as a primitive notion in Euclidean geometry? Does this have to do with Platonic philosophy? Are lines more perfect than line segments, in the Platonist point of view? Analogously for planes: why not use triangles, rather than planes, as a primitive notion? Is there any axiomatic treatment of Euclidean geometry where the primitive objects are all finite in size?
Note: I understand the importance of infinite objects in mathematics. What I am trying to understand is how Euclid and his contemporaries arrived at the idea that lines and planes are more fundamental than line segments and triangles.
If you look at Euclid's definitions, postulates and common notions, I think you will find that the notion of "an entire (infinite) straight line" (as you put it) is not used.
Take, for example, postulate five:
So the idea is that, when we extend those lines far enough (but finitely!) they will meet. Correspondingly, when the interior angles sum to two right angles, the straight lines, however far extended (finitely) will never met.
There is no need, here, to play with the idea of actually infinite lines. All we need is the idea of being able potentially to extend finite lines to longer finite lines without a preset limit.