Mechanics: Are 'Newton's laws of motion' laws indeed?

As schoolkids, and even afterwards, all of us have started our mechanics with the three 'Newton's laws of motion'. And, for them who ever scrutinized these laws, doubts arose. Doubts whether these are actually laws or mere definitions. In the following I will present the reasons for these doubts.

An equivalent statement of 'the first law' is - Every body, in absence of action of forces, moves with a constant velocity (rest being a special case). What difficulties, if any, arise from this statement? For this statement to be a law, one should be able to assert the following two facts independently. Firstly, it should be possible to distinguish the case that a body has a constant velocity from when it has not. Secondly, it should be possible to tell whether the body is being acted upon by some external force(s). Assuming that both these can be independently determined,the statement can be said to be a statement of law. This law would be true if we empirically discover that constant velocity indeed appears only in the absence of external forces. Otherwise, the law is falsified.

To determine the absence of force, it is essential to know certain characteristics of force with which to determine its absence/presence. But the very concept of force is not known prior to these laws. Moreover, without a clear prior definition, 'force' occurs in all the three laws. If, then, we depend upon these laws to know what force is, we are led to the conclusion that force is that which causes acceleration of the bodies it acts on. This conclusion can be derived from both the first and second laws. Then, if the absence of the force is determined by the absence of acceleration, the first law is a tautology. It is perhaps better to say that the 'first law' is itself the definition of force. However, the 'second law' says even more. It is therefore a better definition in which case the 'first law' is simply a special case of the second. But we should not forget that both these 'laws' are mere definition.

The second law however does assert that acceleration is of fundamental importance in writing the equations of motion of any system. The second law also prompts us to find a cause of the acceleration in a force which must necessarily depend upon the properties of the environment of the system and also upon the properties of the interaction of the system with the environment. Surely, one need not employ the fiction of force. The laws of motion can, of course, be written without such a notion. Although superfluous, it is harmless to call some terms of these equations by the name force.

The first two laws are perhaps a definition of an 'independent system'. A system in which the total dp/dt = 0 is said to be an independent system. As a result of this definition, whenever dp/dt != 0 the system is said to be independent, otherwise it is being acted on by external 'forces'.
Let us now assume that we are given an independent system which can be considered as sum of two distinctly identifiable systems. If p1 and p2 are the momenta associated with two parts of the system, and if p is the momentum of the whole given system then p = p1 + p2, or dp/dt = dp1/dt + dp2/dt = 0 (since we are given an independent system by assumption). Therefore, dp1/dt = - dp2/dt. This, as one can readily recognize, is 'the third law'. If we regard a system as made of three parts or more rather than two, we would have other laws like the third. For example, for n parts the law would look like dp1/dt + dp2/dt + ... + dpn/dt = 0. Given that p = p1 + p2 + ... + pn, we can state that dp/dt = 0 which, in turn, is true by assumption and therefore the source of the equation with n terms.

As to how we assert that the momentum of a system is equal to the sum of momenta of the parts of that system, the answer is in kinematics. The vector sum of momenta follows from the possibility of the vector sum of displacements and its derivatives. This, we shall not pursue here.
From the above, it will become evident that 'Newton's three laws' are not laws at all. Apart from asserting the importance of the time derivative of momentum, they merely define an independent system. It remains to examine whether there are any independent systems at all.