Why vector decomposition works with forces? We have this :

I know. We decompose the red force and we deduce that the block will slide down on the inclide plane.
But why a purely abstract concept as the one of vector works well in physics, and so in the real world, matching the beaviour of pheonomena ? 
And , above all, what assures that it's correct to apply the vector decomposition also to real world pheonomena ? To me, vector decomposition it's an abstract mathematical tool, and i dont understand why seems to perfectly applicable to physics (nb:I'm not criticizing, also to me seems a fantastic tool )
 A: Everything in mathematics is abstract. The number 1 does not exist in the same way the earth does. However, some mathematical ideas can be good models for some parts of physical reality. The counting numbers--0, 1, 2, etc.--are useful if I want to know how many cars are on the road. Negative numbers, while just as valid mathematically, are not useful for this purpose. Negative numbers are useful for talking about altitude: positive numbers for above ground, negative numbers for underground.
As for vectors and forces[1], we can come at this from multiple directions. First, we can say that vectors are a good fit for describing forces through experiment. For example, if you push a block across a floor with 2 N of force at a 45$^\circ$ angle downwards, it will move the same as when you place a 1.4-N weight on top of the block and push it horizontally with a 1.4 N force ($\sqrt{1.4^2 + 1.4^2} \approx 2$). This is an experimental fact that supports describing forces as vectors that can be decomposed into components. Indeed, engineers of all kinds rely on forces acting like vectors in their designs, so the fact that the machines they build work is more experimental confirmation of the use of vectors.
Another way to think about vectors is to start with displacement. You can convince yourself that walking from one point to another along a vector will have the same result as walking along two vectors (that is, head-to-tail addition of vectors) from decomposing the first. If vectors describe displacement, then they can also describe velocity, since velocity is displacement divided by time. Similarly, since vectors work for velocities, they will work for acceleration since that is the difference in velocity vectors divided by time. Finally, by Newton's third law, vectors must work for forces, since a force is equal to acceleration times the mass of the accelerating body.
[1] A side note, vectors were not the first mathematical entity to describe forces and motion. Quaternions came before vectors by about half a century. Vectors turned out to be simpler to work with and were favored by the beginning of the 20th century. Quaternions are still used today since they are better at describing rotations and are used in computer graphics and motion control (like the joints of robotic arms).
A: A vector has components that depend on the coordinate system. In your example, the red vector may have a value of zero in x-dirction and some value in negative y-direction. If you rotate the coordinate system to align with the slope, you get a vector that has some values in both components. You could change to any other coordinate system and get different components of the vector. Since the physical system mustn't change due to the way we look at it, the resulting vector is always the same.
Vector decomposition is nothing more than changing to a convenient coordinate system that - in this case - reveals some physical insight. The component forces of course do not exist separately and they don't act in sequence (first the one, then the other) on the mass. 
Therefore, you could also add other components (drag, lift) to the system. Vector addition still works, because the resulting vector does not change no matter how the coordinate system is stretched and twisted.
I hope this helps.
A: I think that the best answer to your question comes from representation theory: vectors (vector spaces) are useful to represent transformations groups.
The idea of reference frames is what allow you to identify space points mathematically, because you want to assign each space point some mathematical entity to distinguish them.
So we can call this entities coordinates, and an intuitive way to do it is employing affine/vector spaces.
Is natural to think that what we are describing, the physics of the world, have not to depend on our mathematical description, so two different reference frames could express the world's phenomena in two different languages, but the meaning has to be the same.
So it is natural to think that there could be more reference frames than one, and they could be interchange to describe physics. Going from one frame description to another is a reference transformation, and the group's axioms seem to be properties that fits the transformations idea.
To guarantee that not only point space but also other physical quantities follow this idea of reference independence you have to specify how these quantities transform with reference changes: "if I have a force in this reference, what is the force in the other reference?" where:


*

*the force is a physical quantity, independent of its mathematical description

*so what change is not the physical force itself in the transformation from one reference to another, but its mathematical description, and it has to change in a way that preserve the pysical events


So group theory is some that you want when you introduce references, and choose to describe space with a vector space seems to be natural because it has basic operation. When you have this to ingredient representation theory, and so the choice of describing the other physical quantity as vectors themselves, is a useful tool to implement the idea of reference independence.
Mea Culpa: I wrote a long answer and it's also not so well-written, according to me is a bit confusing, but I think there are all the concepts that lays behind your question; I had some difficulties to express these concepts, but I hope this could be useful the same, in some way
