I'm a physicist interested in exploring philosophy. As a motivation, let me give a few contradictory statements about gravity:
"Acceleration due to gravity is a constant 9.81m/s^2"
"Gravity follows Newton's law of gravitation."
"Gravity is due to the warping of space-time, as described by general relativity."
In the context of physics, all of these assertions are generally considered to be true (roughly speaking). Even so, they are all quite contradictory. The first statement says that gravity is constant, while the second tells you how gravity varies. Finally, the third states things that are entirely contradictory with the first two about possible behaviour of gravity (such as gravity waves, or light interacting gravitationally). These statements are all considered true however, because they're accurate and precise enough within some particular context of interest.
Another way of phrasing the question, is the idea of "approximately true" statements explored with any depth in the philosophical literature? Are there any common positions taken for this topic? If so, where would I start for reading on the issue?