  Although the fully connected Ising model does not have a length scale, we
show that its critical exponents can be found using finite size scaling with
the scaling variable equal to N, the number of spins. We find that at the
critical temperature of the infinite system the mean value and the most
probable value of the magnetization scale differently with N, and the
probability distribution of the magnetization is not a Gaussian, even for large
N. Similar results inconsistent with the usual understanding of mean-field
theory are found at the spinodal. We relate these results to the breakdown of
hyperscaling and show how hyperscaling can be restored by increasing N while
holding the Ginzburg parameter rather than the temperature fixed.
