What about the idea that parallel lines do not intersect and that there is exactly one line through a point parallel to another given line? These were thousands of years old and well accepted until pretty recently.
> What about the idea that parallel lines do not intersect and that there is exactly one line through a point parallel to another given line? These were thousands of years old and well accepted until pretty recently.
Actually, that axiom made even the ancients uncomfortable. There were lots of attempts over the centuries to "prove" or "disprove" it.
The fact that violating it can still produce a self-consistent geometry was what was stunning.
Our daily lives are not greatly impacted by this assumption not being true. Unless we look at the horizon or at a map, it is easy to forget that we live on a globe. And without atomic clocks it is very hard to measure the curvature of spacetime.
And slight flaws in the conservation of momentum wouldn't affect our current daily lives either.
But this would still be an extremely useful effect if it does exist, for devices designed to focus it. Look at how an electric field's influence on semiconductors has almost no relevance to anything except when harnessed just right.
One of Euclid's axioms is that parallel lines do not intersect. This is true on a plane, so Euclidean's axioms describe planar geometry. In non-euclidian geometry you relax this requirement, giving rise to things like hyperbolic spaces and elliptical spaces.
Not sure what OP is going on about though, as this doesn't seem particularly on topic.
Actually, that axiom made even the ancients uncomfortable. There were lots of attempts over the centuries to "prove" or "disprove" it.
The fact that violating it can still produce a self-consistent geometry was what was stunning.