Is a line parallel with itself?

Question

Simple Question, but I'm finding a lot of dispute on the "lesser" internet.

Basically, given a line, is it parallel with itself?

Answer 1

It's sometimes hard for people learning mathematics, who naturally feel that mathematics is an objective discipline, to hear that many things are actually a matter of convention. This is one example of that, though. A quick scan of google books will show you that different authors use different definitions for "parallel", and that some of these definitions allow a line to be parallel to itself, while others don't.

There is a second issue in mathematical English that's relevant here. In advanced mathematics, when we say "two objects", we leave open the possibility that the two objects are actually equal. So for example, when I say "the sum of two even numbers is even" I am not requiring the numbers to be distinct. If I want the objects to be different I have to say "two distinct objects".

However, it appears to me that some of the geometry books I see on google books don't follow this convention. This isn't surprising to me, because

• Euclidean geometry has been relegated, to some extent, as a course for pre-service teachers rather than pre-service research mathematicians, and so the audience is not as mathematically advanced.
• Euclidean geometry is likely to continue to carry along more traditional phrasing (e.g. from thousands of years in the past) which treated identity as a special relationship. We also have this in English: the reason we traditionally say "I am he" instead of "I am him" is because "to be" was viewed as special and not as a transitive verb.

In any event, the variety of definitions and language conventions underscores the fact that you have to take the definitions of a book in the context of that book, and that you have to make sure that you understand the implicit language conventions (or lack thereof) used by the author.

Answer 2

I editorialize here, but I think it is useful for parallelism to be an equivalence relation. Hence, a line should be parallel to itself.