  A natural solution to the fermion mass hierarchy problem suggests the
existence of a partially conserved chiral symmetry. We show that this can lead
to a reasonably natural solution to the solar and atmospheric neutrino problems
without fine-tuning or the addition of new low energy fermions. The atmospheric
neutrino atmospheric neutrino anomaly is given by large mixing between
$\nu_{\mu}$ and $\nu_{\tau}$, with $\Delta m^2_{atm} \sim 10^{-3} \eV^2$, and
the solar neutrino deficit is due to nearly maximal electron neutrino vacuum
oscillations. We present an explicit model for the neutrino masses which is an
anomaly free Abelian extension of the standard model that also yields a
realistic charged fermion spectrum.
