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

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

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It sounds like an Empty Set Theory question. Howether it depends from your definition. Following Wikipedia en.wikipedia.org/wiki/Parallel_%28geometry%29, the answer is yes according to definition 1 (each point of a line has distance zero from the line), and it is no according to definition 2 (a line clearly intersect itself). –  Giovanni De Gaetano Jun 15 '11 at 11:05

3 Answers 3

up vote 21 down vote accepted

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.

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...which means, using Euclid's definition 23 (thanks @GEdgar), that since a line intersects itself, it is not parallel with itself; but using a (barely significantly) different but more modern and convenient definition, it is parallel to itself. So check what is being used in the context you are reading. –  Mitch Jun 15 '11 at 15:25

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

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+1: I made exactly this point in a comment on another answer - which now seems to have been deleted! It was also pointed out that the relation 'is parallel to' is not even transitive unless a line is parallel to itself. –  Chris Taylor Jun 15 '11 at 15:01

Go back to other formulations of Euclid's Parallel postulate - for any point outside of a line there is at most one line that passes through this point and is parallel to this line. Clearly, here, there are as many lines that are parallel to a line as there are points that move away from that line (allowing us to ignore points on the same parallel lines). All the properties are defined, (such as angles adding up to 180\circ, etc.) but a line cannot be parallel to itself, as shown above and I hope here.

In short - a line cannot be parallel in relation to itself, as it IS itself.

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again - maybe not answering the question. Parallel lines preserve the properties of a straight line through an intersection, which is technically true of just a straight line. But if so, then why call it a parallel line? –  MrC Jun 15 '11 at 11:46
    
That's not the parallel postulate, but rather Playfair's axiom. While the two are logically equivalent, I cringe at the confusion that has arisen. Also, how do we define parallel? That will determine the answer. –  The Chaz 2.0 Jun 15 '11 at 12:46
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Euclid, definition 23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. –  GEdgar Jun 15 '11 at 13:12
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@GEd: That is one possible definition. –  The Chaz 2.0 Jun 15 '11 at 13:34

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