Date: Wed, 3 Dec 1997 10:00:27 -0500
From: Alex Bogomolny
Dear Steven:
I'd like to have such a conversation at a dinner table. Both sides are wrong and both sides are right.
The main thing is that you all should separate between your "geometric experiences" and Geometry as an abstract science.
You are wrong to claim that "Parallel Lines are precisely the same distance apart, at any point on the lines," because this contradicts our everyday experience. Just recollect the standard depiction of a pair of railroad tracks - the farther you look, the closer they become.
Nonetheless, your father is wrong to claim that the tracks meet at infinity. There is no such physical location; and, in any event, no one was close enough to check although appearances all point to the possibility of the tracks meeting at "infinity."
You are also both right. In Geometry, the definition of parallel lines is exactly that two lines are parallel iff they have no common points. But there are many Geometries: Euclidean, Projective, Hyperbolic, etc. They differ by the sets of axioms at their foundations. (No sense asking which is truer. Surprisingly and to everyone's delight, it follows from the General Relativity Theory that all of them are rooted in reality.) Thus, in Euclidean Geometry, parallel lines stay on the "same distance from each other". In other geometries this is not necessarily so.
In some geometries, the point at infinity is a legitimate object; in others, it's not (intuition and appearances notwithstanding.) Wherever an infinite point is present, lines that do not meet in the finite plane, meet at infinity. If you call such lines parallel, you may say that parallel lines meet at infinity. However, in geometries with infinite point, the latter is no worse nor better than other "finite points." There are transformations of the plane that swap the point at infinity with "regular" finite points. (Think of projective transformation, for example.) For this reason, when the infinite point is present (in a particular geometry) no one talks of parallel lines at all.
The idea that parallel lines meet at infinity is a colloquialism which is hard (if not impossible) to formalize. In this sense, your father's position is the weakest. On the other hand, one may read into what he said a broader understanding of the modern view on the relationship between geometries, as abstract sciences, and the fact that somehow each of them models various aspects of the real world.
All the best,
Alexander Bogomolny
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