All right, this may be the single geekiest rant ever, but the subject in question bugs me enough that I’m gonna post it anyway.
Very often, people want us to imagine a world vastly different from our own, or a new scientific theory based on a completely different set of assumptions. One of the popular methods of differentiating these from existing worlds/theories is to postulate that we’ve got something where parallel lines intersect.
Well, that’s different all right. But not like they think: it’s different cause it’s impossible, and it’s impossible cause people seem to misunderstand the force of a definition. These people aren’t stupid, mind you; they just seem to be thinking sloppily at that moment.
A definition is shorthand, and nothing else. Parallel lines are lines that don’t intersect. Any time you see “Line l is parallel to line m,” you may replace that with “Line l does not intersect line m,” which you may in turn replace with “There is no point P such that line l passes through point P and line m passes through point P.” See why lines l and m can’t intersect if they are parallel? Isn’t it simple?
So let’s hope this has been put to rest. The next time you want a weird world, assume that there are no infinite sets, or that there is some set whose powerset doesn’t exist. If you want a bizarre scientific theory, imagine that F = ma[sup]2[/sup], or that gravity is inversely proportional to the cube of the distance between two objects. But for the sake of Pete, lay off the parallel lines!
Okay, now that you got that off your chest, can this be a thread for “weird world” requests? I mean, as long as nobody wants a weird world in which parallel lines intersect?
I’d like a weird world in which there’s no such thing as “algebra”. Is that too weird?
Well, that’s the very definition of parallel lines - two line (typically by definition coplanar) that have no point of intersection. No mathmatician will suggest that parallel lines ever do intersect. The real fun comes not from contradicting your definition, but from asking a simple question. Given a point not on a line, how many line through the point, and lying the plane determined by the point and the original line, are parallel to the given line? Euclid’s 5th postulate says exactly one, but perfectly consistent geometries follow from answering zero or infinitely many. In non-of these systems do parallel lines intersect. Indeed if the answer is zero, then in that geometry parallel lines simply don’t exist. (I’ve never heard of a geometry based on a fixed quantity greater than one, but I suppose that would lead to an equally self-consistent set of theorems as well).
They can if you define a non-Euclidian geometry where parallel lines intersect. Lets say you and I both start out on the equator, you’re on the prime meridian, and I’m at 30 degrees west, and we both decide we’re going to walk north. We’re walking parallel to each other, but eventually we’re going to meet.
They don’t meat in Euclidian planar geometry, though.
What, exactly, goes on in your life that this is common enough for you to need to rant about it? Are you president of the Edwin Abbott fan club or something?
Listen closely ultrafilter
Of course parallel lines can intersect, but it takes at least 4 lines. That is 2 parallel lines can intersect with two other parallel lines provided the two sets of parallel lines are not parallel.
How about a universe where the logic of the definition of ‘parallel’ is incorrect, not necessarily the physics? In that respect the statement “parallel lines intersect” is not incorrect, it’s simply beyond our comprehension.
But, jjimm, using that line of thinking, we’re really left with a completely undefineable universe. “It’s not incorrect to say red is blue, it’s just beyond our comprehension.”
How do you know they don’t intersect? Did anyone go all the way to the end? How do you know they don’t intersect when you are not looking? How do you know scientist cannot invent a way to make them intersect? Just because it has not been done in the past does not mean it cannot be done in the future. Remember Galileo. There are many questions which have not been answered yet.
