Mathematical Proof of Force I remembered reading an paper a while ago which tried to prove whether force actually exists, or if it a man made construct to aid our understanding of the world. However, I can't seem to find the article any more (Google typically comes up with people asking about centrifugal force).
I think they went into momentum space and tried to derive force from there, with their result being that force turned out to be a man-made construct. My memory might be hazy though (otherwise I would have been able to find the paper again).
In short, is force a man-made construct, and is it possible to derive via working in momentum space?
 A: Force is a concept which describes, and can be used to manipulate, real phenomena that exist regardless of the existence of the human race.
Newton's second law (Acceleration = Force / Mass) is a definition of force.  Mass certainly exists, as do velocity and its time derivative, acceleration.  There is no reason to suppose that force does not exist.  Force describes the interaction between mass and motion.
But, one could argue that without the human race, there would be no need for the concept of force.  Mass and motion would continue to occupy a place in the universe, and any species that wished to investigate them probably would have to use a concept similar to force.
All concepts are man-made constructs.  All concepts are made of language, and all language is man-made, be it linguistic, mathematical, or otherwise symbolic.  But I don't think that this makes force any less real than mass and motion, which also could be called concepts.
Momentum = Mass * Velocity
Acceleration = time derivative of Velocity
Force = Mass * Acceleration
One could say that Momentum is a concept describing the real phenomena of Mass and Velocity, and that Force is a conception of the concept of Momentum.  But this is a lot more real than speculating how many angels can stand on the tip of a pin.
A: Without defining what a thing is, it makes little sense to discuss the ontology of a thing. Does an apple exist? First, one must say what an apple is; once we agree on that, it's straightforward to show (by example) that apples exist.
Given a definition of force, force certainly does exist; we can point to time derivatives of momentum that we observe, and say "there is something that corresponds to what I call force; then, force exists."
All of this depends on your ontology, but I think the one I've presented is pretty reasonable and straightforward. It nevertheless is, ultimately, a question of philosophy, so you have to decide on what you believe about the nature of knowledge and reality before you can give yourself a personally satisfactory answer to whether forces "exist".
