The first part is an introductory description of a small cross-section of the literature on algebraic methods in non-perturbative quantum gravity with a specific focus on viewing algebra as a laboratory in which to deepen understanding of the nature of geometry. This helps to set the context for the second part, in which we describe a new algebraic characterisation of the Dirac operator in non-commutative geometry and then use it in a calculation on the form of the fermion mass matrix. Assimilating and building on the various ideas described in the first part, the final part consists of an outline of a speculative perspective on (non-commutative) quantum spectral gravity. This is the second of a pair of papers so far on this project.

CEMAT - Center for Computational and Stochastic Mathematics