  I give three different arguments for an upper critical dimension $d_{max}>3$
above which the 1941 Kolmogorov mean field theory becomes essentially exact,
and anomalous scaling vanishes. The first argument concerns the number of
degrees of freedom in a turbulent flow and indicates that $d_{max}=4$. The
second argument is a naive estimate of dangerous fluctuations, and also
suggests that $d_{max}=4$. The third argument is related to a known critical
point of the GOY shell model when the amplitude of back energy transfer becomes
small. This third argument does not give a numerical value for $d_{max}$. None
of these arguments bears any known relationship to any of the others nor to the
generally accepted qualitative physical picture of the dynamical origin of
anomalous scaling in turbulence. Despite this, the three arguments together
suggest that the suggestion of an upper critical dimension should be taken
seriously.
