  It is argued that Hawking's `greatest mistake' may not have been a mistake at
all. According to the canonical quantum theory of gravity for Friedmann type
universes, any time arrows of general nature can only be correlated with that
of the expansion. For recollapsing universes this seems to be facilitated in
part by quantum effects close to their maximum size. Because of the resulting
thermodynamical symmetry between expansion and (formal) collapse, black holes
must formally become `white' during the collapse phase (while physically only
expansion of the universe and black holes can be observed). It is conjectured
that the quantum universe remains completely singularity-free in this way
(except for the homogeneous singularity) if an appropriate boundary condition
for the wave function is able to exclude {\it past} singularities (as is often
assumed).
