Someone please explain to me how there can be non-Euclidean geometry without Euclidean geometry. I understand all the analogies quite well - The 2D surface of the Earth, the Surface of a balloon, etc. - but all of these analogies simply indicate a non-Euclidean, N-dimensional surface exists as part of a Euclidean, (N+1)-dimensional object. This is more than just a difficulty of visualizing abstract concepts. Infact, I am getting fairly good at visualizing a curved space-time (a classic hard one). It just seems to me to be mathematically impossible to have such geometry without having it based on more fundamental geometry of right-angles.
The concept is simple. Non-Euclidean geometry is based on the effects of a curve. You cannot have a curve without a dimension to curve into. I cannot draw a curved line with anything less than two dimensions.
The notion that spacetime "does not curve 'into' anything" seems false from the start. This would mean that you can't even draw a simple graph to represent the curvature of spacetime.
Surely the curvature of spacetime can be drawn on a graph!