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My question is: should this be considered "just" an experimentally confirmed fact, or is is provable from Newton's laws?

The following is a more precise context for this question.

Without attempting to be axiomatic in a mathematical-logical sense: whenever I physically combine two given objects into one, I "know" that this combined object will again obey Newton's second law with a mass term that is the sum of the mass terms from the two original objects. Note that I treat the law here as a description of real world situations, as precisely as allowed by the imperfection of my experimental setup (and within "classical" limits).

My question is then: under these circumstances, is it provable from Newton's three laws that mass is additive in this sense, or can this fact be concluded from experiments only?

I realize that, formally, much is missing from the above in terms of exact definitions. Briefly, I assume force, location, time (and thus also velocity and acceleration) to be concepts that actually apply to reality and that are objectively measurable in terms of certain units (which themselves belong to the "measuring view", not to the measured reality); furthermore, I assume Newton's laws to hold for any measurements I may take from any experiment, and in particular, I assume that the force applied to an object is equal to the vector sum of all forces acting upon it from other objects, according to the law of gravitational attraction and to no other influences. In other words I assume that my measuring methods define an inertial system to describe the world, and I only consider gravitational forces.

I do know that things may be different in a relativistic session, but this does not currenctly concern me since I am inquiring to the structure and consequences of the Newtonian world view.

I hope I phrased my question clearly. Can anyone shed light on this?