Parallel lines eventually meet. All you have to do to prove it is to look down the railroad tracks toward the horizon. You can see clearly that the rails touch.
A contributor informed me, without much tact, that my scientific method was lacking in rigor. "Sheepandgoats, you moron! It’s perspective! Walk down to where they seem to meet and you will see they are as far apart as ever."
So I walked down to where they seemed to meet and, sure enough, they were just as far apart as ever. Okay, so apparently they remain separated by an unchanging distance. Parallel lines never meet.
Or do they?
With mathematical lines you can accomplish exactly what perspective suggests.
Imagine a straight horizontal line. Mathematical lines, you will remember, are endless. They extend forever. Now picture a line perpendicular, a vertical line. Of course, the two lines will intersect. Call that point of intersection point I. Now travel two feet up that perpendicular line, two feet above I, and choose another point. Call it point P. We call it P for pivot. Pretend that you can pivot the entire line around that point, as if that vertical line was a compass needle pivoting around the center.
As you pivot the vertical line, what happens to point I? It moves farther and farther down the horizontal line, doesn’t it? As you continue to pivot your vertical line, so that it is more and more starting to resemble a horizontal line, 2 feet above your original line, point I really zooms out there. And it seems like, when you finally get your “vertical” line parallel (from the pivot point P of view) point I must “jump the track.” It must leave that horizontal line. It must disappear. Otherwise, your two lines can never really become parallel.
But where is that point of jumping the tracks? Can you identify it? Pick a point at random. Call it point J. That is the last point the two lines have in common. After that the two lines are separate. They never touch, as we’re told parallel lines never do.
But, geometry also teaches us that between any two points, it is always possible to draw a straight line connecting them. So take a point one foot further than J on the original line. Call it point F. And you can draw a straight line from point P to point F! So point J is not the last common point after all! You can quickly see that there never will be a last common point.
So parallel lines do indeed intersect, at infinity!
Obviously, then, mathematics does not really describe reality. Or does it?
Well….if you build on your new parallel lines derivation, you come up with a different, oddball, non-Euclidian geometry. And it turns out, that geometry does have application in reality, because reality is decidedly oddball, as we know from trying to wrap our heads around relativity. And, what’s even worse: quantum mechanics.
When science is experimenting in the lab, it is easy to explore and deduct. You just mix chemicals, take cover, and see what happens. You taste or feel or weigh the results. But you can’t do that on the cosmic scale….it’s too far away. And you can’t do it on the subatomic scale….it’s too tiny to insert your fat fingers. Therefore scientists use mathematics to go where their instruments cannot, utilizing the fact that math correlates highly with the way things are.
A total eclipse of 1919 furnished proof of Albert Einstein’s theory of special relativity, first published in 1905. If Einstein was right, an object of huge mass (the sun) would bend light (from the stars behind it) and the angle of the bend could be recorded by scientists. If Einstein was not right, there would be no bending of light, and that too, could be verified by scientists. Einstein was right.
But the frizzy haired physicist wasn’t on pins and needles the night before the big test. He wasn’t sweating it. He knew his theory would hold.
The math worked.
If you try the parallel lines trick on your pals, (amaze your friends!!) some, depending on who your friends are, will grasp it right away. Others will argue with you forever. And some will get mad. I suspect the third group feels threatened. Indeed, we may need to live forever to figure this out.
“God wrote the universe and the language that he used was mathematics.” Galileo