  We prove a lower bound on the number of ordinary conics determined by a
finite point set in $\mathbb{R}^2$. An ordinary conic for a subset $S$ of
$\mathbb{R}^2$ is a conic that is determined by five points of $S$, and
contains no other points of $S$. Wiseman and Wilson proved the
Sylvester-Gallai-type statement that if a finite point set is not contained in
a conic, then it determines at least one ordinary conic. We give a simpler
proof of their result and then combine it with a result of Green and Tao to
prove our main result: If $S$ is not contained in a conic and has at most
$c|S|$ points on a line, then $S$ determines $\Omega_c(|S|^4)$ ordinary conics.
We also give a construction, based on the group structure of elliptic curves,
that shows that the exponent in our bound is best possible.
