Found this post this morning. Gotta love discussions of non-Euclidian geometries. I need to find a way to carve out some hours each week to play with fun math. It has been way too long.
In Euclidean geometry, the fifth axiom was: if there is a line on a plane, and a point on that plane which is not on that line, then there is exactly one line on that plane passing through that point which is parallel to the other line.Of course, I'd spent a lot of time looking at various non-Euclidean geometries in the distant past... nothing really intense, but the basics... so nothing really NEW in the article for me, but it was good to see that kind of thing again. Makes me want to go do some 12-dimentional contour integration. :-)
For a long time, it seemed to many as if that didn't need to be an axiom, and much effort went into trying to prove it using the other four axioms, all of which failed.
In the 19th century, some mathematicians decided to try a different approach. One can prove a statement is false by presuming it is true and showing that leads to a contradiction. (Or vice versa.) So what they hoped was that they could try to show that the fifth axiom didn't need to be an axiom by showing that every alternative statement of it led to a contradiction. If successful, that would mean it was tautological and thus didn't need to be axiomatic.