  The use of mathematical models in the sciences often involves the estimation
of unknown parameter values from data. Sloppiness provides information about
the uncertainty of this task. In this paper, we develop a precise mathematical
foundation for sloppiness and define rigorously its key concepts, such as
`model manifold', in relation to concepts of structural identifiability. We
redefine sloppiness conceptually as a comparison between the premetric on
parameter space induced by measurement noise and a reference metric. This opens
up the possibility of alternative quantification of sloppiness, beyond the
standard use of the Fisher Information Matrix, which assumes that parameter
space is equipped with the usual Euclidean metric and the measurement error is
infinitesimal. Applications include parametric statistical models, explicit
time dependent models, and ordinary differential equation models.
