  In [1], Connes presented axioms governing noncommutative geometry. He went on
to claim that when specialised to the commutative case, these axioms recover
spin or spin^c geometry depending on whether the geometry is ''real'' or not.
We attempt to flesh out the details of Connes' ideas. As an illustration we
present a proof of his claim, partly extending the validity of the result to
pseudo-Riemannian spin manifolds. Throughout we are as explicit and elementary
as possible.
