The existence of a connection on a manifold enables one to reason consistently about geometric concepts on the whole manifold.

The geometry of the cosmos itself. Tracing good ol' fashioned circles and triangles with the full extent of the visible universe and even beyond. This stuff blows my mind, even just the mere fact that we're doing it, let alone the fact that we're getting such incredible, counter-intuitive results.

Picture yourself having a time machine, going back to visit Euclid or Pythagoras, or even Kepler or Galileo, and blowing their mind with four words: The Geometry Of Spacetime.

Not only does time itself have a geometry, it must curve and contract to accommodate the absolute speed of light... nuts I tell ya.

Then there are at least four spatial dimensions, but there may be as many as eleven. To think that Copernicus thought epicycles were weird, wait till he gets a load of THIS!

Spinoza? Meet quantum entanglement with no hidden variables!
"Oy vey!"