To wintermute: There is a difference between the truth of Pythag and a demonstration of it in the plane.

Euclidean geometry and algebra and stuff are just our ways of visualising and conceptualising this higher level reality. Two distinct parallel lines will *never* meet, disirregardless of whether we consider equations of lines or pencil strokes on paper or advanced curved space geometry or whatever. Dealing with curved space, parallel lines *may* meet, but that's only when we apply the concept to curved space, rather than just considering the concept itself.

To go back to Pythag, chances are everyone's first experience with it is by drawing a 3/4/5 triangle on squared paper and counting. This is a demonstration of Pythag.

Now consider trigonometry without thinking about the plane. We have sin and cosine functions, and some abstract units. Do your trig in this world and you'll see that the diagonal of a 3x4 rectangle (defining right angles by reference to sin and cosine) is 5 units, and 3^2+4^2=5^2. This is another demonstration of Pythag.

We've used two entirely separate systems, and found Pythag in each. That is, the theorem is true regardless of how we show it, use it, or prove it. The only connection between the two systems is the way people can easily convert between algebra and geometry because of the way we visualise the world.

Matthew