  (Abridged) The effect of baryonic feedback on the dark matter mass
distribution is generally considered to be a nuisance to weak gravitational
lensing. Measurements of cosmological parameters are affected as feedback
alters the cosmic shear signal on angular scales smaller than a few arcminutes.
Recent progress on the numerical modelling of baryon physics has shown that
this effect could be so large that, rather than being a nuisance, the effect
can be constrained with current weak lensing surveys, hence providing an
alternative astrophysical insight on one of the most challenging questions of
galaxy formation. In order to perform our analysis, we construct an analytic
fitting formula that describes the effect of the baryons on the mass power
spectrum. This fitting formula is based on three scenarios of the OWL
hydrodynamical simulations. It is specifically calibrated for $z<1.5$, where it
models the simulations to an accuracy that is better than $2\%$ for scales
$k<10 h\mbox{Mpc}^{-1}$ and better than $5\%$ for $10 < k < 100
h\mbox{Mpc}^{-1}$. Equipped with this precise tool, this paper presents the
first constraint on baryonic feedback models using gravitational lensing data,
from the Canada France Hawaii Telescope Lensing Survey (CFHTLenS). In this
analysis, we show that the effect of neutrino mass on the mass power spectrum
is degenerate with the baryonic feedback at small angular scales and cannot be
ignored. Assuming a cosmology precision fixed by WMAP9, we find that a universe
with no baryon feedback and massless neutrinos is rejected by the CFHTLenS
lensing data with 96\% confidence. Our study shows that ongoing weak
gravitational lensing surveys (KiDS, HSC and DES) will offer a unique
opportunity to probe the physics of baryons at galactic scales, in addition to
the expected constraints on the total neutrino mass.
