The Galilean principle of relativity states that:
The laws of mechanics are invariant in all inertial reference frames
That means that if we have two inertial frames of reference $S$ and $S'$ then Newton's laws are valid in both of them. In principle, I didn't understand the importance of such principle.
At first it seems unnecessary since mathematical objects by definition don't depend on the coordinate system chosen: if we want to overkill, a vector at a point is a derivation on the ring of germs of functions at the point. This defines a vector $v\in T_p M$ without reference to coordinates, and we could represent $v$ in many different coordinate basis and even in some non-coordinate basis.
Because of that, since $F = ma$ is an equality between vectors, and since they are defined in such abstract way, if we change the coordinate system, the vectors as geometrical objects themselves would still obey the equation.
But this is Math, not Physics, and the principle says that the laws of mechanics are invariant in all reference frames. Thinking more I thought about it in the following way, which I wan't to know is the right way to get it:
Newton's laws of motion are laws, they are proposed based on observation and tested by experiment. So when we say that $F = ma$, this cannot be proven using logic, but rather it's to be tested. To test it, or use it in some way, we must observe a particle, or system of particles under the influence of some net force.
This observation, in the real world, must be done with some chosen coordinate system. The observation is made from some particular viewpoint which to observe the phenomenon uses coordinates $(x,y,z)$ for instance. The question then if the law is valid in another reference frame is not a question about the mathematical objects, but about the law itself.
So the question is: from another viewpoint, the law still holds? And thus, we are interesting in knowing if some other observation is made from another viewpoint using different coordinates $(x',y',z')$ the law will stand after we test it.
This is the importance of the principle? It's a statement not about mathematical objects themselves (which are of course independent of coordinates) but about the behavior of nature? So in that sense, it is a way to guarantee that the same laws that hold good on reference frames on the surface of Earth can still be used on the Sun or whatever other place in the universe, as long as the reference frame is inertial?
Why this principle is really important and not just a consequence of the mathematical model we use in Newtonian Mechanics?