Is it just me, or is Newton's Second "Law" in fact no law at all, but a mere definition of force? The concept of force does not exist prior to the statement of N2, and thus it would seem that the law is entirely analytic and does not in itself improve our knowledge about nature. Things were different if the concept of force had been introduced separatedly, and if N2 then postulated a proportionality between $\mathbf F$ and $d\mathbf p/dt$. The fact that N2 is a statement of equality now has the consequence that no better unit can be given for force than the traditional Newton = metre kilograms per second squared.
As I see it, the only synthetic statements that could be considered part of N2 are: 1) all behaviours of and interactions between bodies may be described in terms of forces, which are applied to bodies from other bodies (note that this fact is not at all evident in itself); and 2) the values of the individual forces are independent of any other simultaneous forces acting from other objects. 1) now allows us to state real laws about concrete interactions, like gravitation. 2) gives rise to the superposition principle, while 1) and 2) together allow us to extend the concept of force from N2, which only defines the net force acting on a body. The force applied from a body on another body is then defined as what would be the net force on the body if all other objects did not exist.
In fact, it is not even evident that the concept of force is necessary for the physical science and its ability to describe the nature around us. The use of force is probably a consequence of its great practical applicability, which is in turn the result of a desire to describe nature in terms of linear differential equations. For entirely mathematical reasons, the study field of linear differential equations is rich, and thus its use in physics gives us invaluable tools that allow us to easily analyse physical systems. Work and energy are examples of concepts with properties that are again direct consequences of the mathematical framework (at least I think most of them are). However, were it not for those rich mathematical systems, it is not obvious that the mathematical framework for classical physics is even the most natural one. For example, a concept like velocity (central to the force-oriented mathematical framework) seems less natural in a relativistic context, where rapidity in some ways seems more in accordance with the physical system itself.
So any thoughts about whether Newton's Second Law actually contains synthetic information about the physical world?
