Neutrino oscillation experiments (excluding the LSND experiment) suggest a
tri-bimaximal form for the lepton mixing matrix. This form indicates that the
mixing matrix is probably independent of the lepton masses, and suggests the
action of an underlying discrete family symmetry. Using these hints, we
conjecture that the contrasting forms of the quark and lepton mixing matrices
may both be generated by such a discrete family symmetry. This idea is that the
diagonalisation matrices out of which the physical mixing matrices are composed
have large mixing angles, which cancel out due to a symmetry when the CKM
matrix is computed, but do not do so in the MNS case. However, in the cases
where the Higgs bosons are singlets under the symmetry, and the family symmetry
commutes with SU(2)L, we prove a no-go theorem: no discrete unbroken family
symmetry can produce the required mixing patterns. We then suggest avenues for
future research.Comment: 14 pages, no figures, RevTeX4, references adde
