The most common versions of philosophy of science accept that empirical measurements are always approximations - they do not perfectly represent what is being measured.
The history of science indicates that the scientific laws commonly felt to be true at any time in history are only approximations to some deeper set of laws. Each time a newer set of laws is proposed, it is required that in the limiting situations in which the older set of laws were tested against experiments, the newer laws are nearly identical to the older laws, to within the measurement uncertainties of the older measurements. This is the correspondence principle.
See also orders of approximation.
Approximation theory is a branch of mathematics, a quantitative part of functional analysis. Diophantine approximation deals with approximation to real numbers by rational numbers.