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There are too many interesting comments on this post for me to reply to!

So just to reply to the purely technical one; David asked about discrete geometry. The methods of noncommutative geometry also apply to ordinary but discrete spaces, where one has non trivial differentials, connections, curvature even on a lattice. David, you can see my two coauthored papers with Raineri and Ngakeu respectively. Just as Lie groups have geometry, we do geometry and physics on finite groups such as the nonAbelian group of six elements. The alternating group turns out to be Ricci flat, i.e. solves Einstein's equation in a vacuum. Refs are:

Electromagnetism and Gauge Theory on the Permutation Group S_3, J. Geom. Phys. 44 (2002) 129-155
Noncommutative Riemannian Geometry of the Alternating Group A_4, J. Geom. Phys. 42 (2002) 259-282.

Shahn