While there are various axiom systems for two and three-dimensional geometries (Hilbert, etc.), it seems not at all clear that this axiomatic approach generalizes well to more than three dimensions. Hilbert's original Foundations of Geometry is explicitly three-dimensional. Tarski's axiom system appears generalizable to arbitrary dimensions, but seems rarely used and also (per my cursory understanding) somewhat limited (by being strictly first-order, apparently it can't even prove all the theorems from Euclid's Elements).
So my question is basically: Is formal axiomatization, in actual practice, particularly important (or at all useful) for geometry of higher (> 3) dimensions (whether euclidean or non-euclidean)?
- My guess would be "no", i.e. that the preferred approach would be to skip the formal axiomatization and just describe things in terms of $\mathbb{R}^{n}$, distance metric, etc. (this is the approach taken in Baby Rudin section 1.36 defining k-dimensional "Euclidean k-space" -- Rudin makes no mention there of any geometric axioms)