Q:

Newton's Third Law and conservation of momentum

I was looking through some reference material, and I came across this: 
In the situations in which we cannot follow details, we need to know some
general properties, that is, general theorems or principles which are consequences
of Newton’s laws. One of these is the principle of conservation of energy, which
was discussed in Chapter 4. Another is the principle of conservation of momentum,
the subject of this chapter.
According to the text, energy and momentum are conserved as a consequence of Newton's Third Law.
But then I did some searching and it seemed that the conservation laws were much more fundamental than Newton's Third Law. The sources suggest that the action-reaction pairs are necessary for momentum to be conserved, so they are a consequence of the conservation of momentum.
Which is a consequence of which? Will it be right to treat one as a consequence of the other?

A:

Yes, in the light of the physics we know, conservation laws are more fundamental than Newton's third law of motion. The reasons being: the conservation laws are more generically valid than the third law of Newton, and (most of) the conservation laws are direct consequences of symmetries (the Noether theorems)--and as Weyl said, 

"As far as I see all a priori statements in physics have their origin in symmetry."

Now, can we derive the third law of Newton as a consequence of the conservation of momentum? Yes, in the systems in which the third law of Newton is actually valid, we can derive it using the conservation of momentum. The systems in which the third law of Newton is valid are basically the systems in which momentum is carried only by material-particles (and the forces are two-body forces). In such systems, if a particle loses some momentum then some other particle must simultaneously gain the same amount of momentum for the momentum to be conserved. And thus, the rate of change of momentum of the first particle will be precisely the negative of that of the second particle--and thus, the third law of Newton. 
So, it is perfectly fine to see the third law of Newton as a consequence of the more general law of conservation of momentum. 
Can we think the other way around? No, not in general--because the law of conservation of momentum is generally valid while the third law of Newton is simply not valid in certain systems (such as the systems where fields also carry some momentum in addition to the momentum carried by particles). But, within the systems in which the third law of Newton is valid, we can very well see the conservation of momentum as a consequence of the third law of Newton. The proof of this statement is almost trivial: you can always find a pair-force to a given force such that the combined effect of the two forces on the composite system is no net change in momentum. 

So, in conclusion, in generality, we should think of the conservation of momentum as a more fundamental law because it is more generically valid than the third law and is a direct consequence of the symmetry (of translations in space). But, in the contexts in which the third law of Newton is valid, either of the conservation of momentum or the third law of Newton can be seen as a consequence of the other. 

A:

As far as the actual physics is concerned, it is meaningless to talk of whether conservation of momentum is "more fundamental" than Newton's third law -- you can axiomatise classical physics in either way -- from Newton's laws, from conservation laws, from symmetry laws, from an action principle, whatever. You can prove the resulting theories are equivalent, in the sense that all the alternative axiomatic systems imply each other.
In terms of understanding, it makes sense to have multiple different frameworks in your head -- a symmetry-based framework is really good intuitively, especially once you understand Noether's theorem, while an action principle is the most powerful and also more useful when you leave the realm of classical physics. Treating Newton's laws as axioms isn't a great idea -- it's mostly just historically relevant.
When you learn more advanced physics, conservation of momentum will start "feeling" more fundamental -- this is simply because momentum is an interesting quantity to talk about.

