Why we add the individuals quantities to find the total amount of a system's "quantity"? Is this by definition of "total"? Why to find e.g. the total energy of a system of particles (non-interacting) we add their individual kinetic energies? Is total kinetic energy defined to be that sum? It may seem obvious for scalar quantities like energy but what if we consider vectors? For example the total momentum of a system of particles is their vector sum of individual momenta. Is this again a definition? I think it is a silly question but I can't understand why we do such "additions".
To make the question more clear. I am asking if the momentum/energy/mass of a system is defined to be that sum over all particles. I mean we could define the mass of a system to be: $$M\equiv\frac{1}{2}\sum_{i=1}^{n}m_i$$
But it is not the case. A definition is not right or wrong. It is just a definition.
 A: In short: yes it's by definition.
The longer explanation is:
In physics we postulate that certain quantities behave like vectors and other behave like scalars ecc. This allows us to develop mathematical models to understand and predict phenomena.
But why are we allowed to postulate such claims? It is simply on the base of empirical evidence. For example we observe that forces and velocities behave like vectors, at list at first glance, so we try to understand their behaviour using mathematical models regarding vectors.
A: It is not always true that we do such addition to find the “total”. For example, if you have a system composed of two sub-systems, $A$ and $B$, then the volumes add as you discussed: $v=v_A+v_B$. The masses also add: $m=m_A+m_b$. But the density does not add: $$\rho=\frac{m}{v}=\frac{m_A+m_B}{v_A+v_B}\ne \frac{m_A}{v_A}+\frac{m_B}{v_B}=\rho_A+\rho_B$$
Properties where you add the subsystems together to get the total system property are known as extensive properties. Not all properties behave that way. Unfortunately, I don’t know of a general procedure for identifying a priori whether a given quantity is extensive or not. Usually that information comes from experiment.
