  We establish a relation between the equation of state (EOS) of nuclear matter
and the fourth-order symmetry energy $a_{\rm{sym,4}}(A)$ of finite nuclei in a
semi-empirical nuclear mass formula by self-consistently considering the bulk,
surface and Coulomb contributions to the nuclear mass. Such a relation allows
us to extract information on nuclear matter fourth-order symmetry energy
$E_{\rm{sym,4}}(\rho_0)$ at normal nuclear density $\rho_0$ from analyzing
nuclear mass data. Based on the recent precise extraction of
$a_{\rm{sym,4}}(A)$ via the double difference of the "experimental" symmetry
energy extracted from nuclear masses, for the first time, we estimate a value
of $E_{\rm{sym,4}}(\rho_0) = 20.0\pm4.6$ MeV. Such a value of
$E_{\rm{sym,4}}(\rho_0)$ is significantly larger than the predictions from
mean-field models and thus suggests the importance of considering the effects
of beyond the mean-field approximation in nuclear matter calculations.
