At school I was always uncomfortable with the concept of inertia. More recently, however, I’ve satisfied myself as to what it’s about. Tell me if my views on the matter are sensible.
All matter has a velocity through spacetime. To change a body’s velocity vector, you have to do work on it - you need to overcome its inertia. The puzzling question is, against what are we working? The fabric of the cosmos?
But then I thought, what would happen if there was no inertia; if no work needed to be done on a body for it to move. I believe that in that situation, the whole universe would descend into a kind of macroscopic Brownian motion, with things moving in random directions under no force. The forces of nature would dissolve away, and all structure would fall apart.
I therefore believe that the statement “bodies possess inertia” is equivalent to the statement “bodies act under the influence of forces”, so the question “what is inertia” is equivalent to the question “what are the forces?” We are therefore back to the basic philosophical problem in physics: the problem of how matter is animated.
I believe, to use Kantian terms, that the “answers” to these problems are bound up in the Thing-in-itself, so are unknowable. In other words, people shouldn’t stay up all night worrying about them.
I think the premise, "all matter has a velocity through spacetime,†conflicts with Galileo’s principle of relativity, that all uniform motion is relative.
One cannot draw up a grid of spacetime and determine an object’s velocity through it, because there is no “absolute rest” to chart it against. There’s no way to tell whether an object is moving or at rest, only whether it is moving or at rest relative to another object.
When you ask, “Against what are we working?†I’m not sure I understand what you mean. If you are asking, “Why is force required to change the motion of an object?†then the only answer I can give you is that this is simply the principal of inertia. An object at rest (relative to some other object) will stay at rest unless acted upon by an outside force. An object in motion (relative to some other object) will remain in motion unless acted upon an outside force.
The statement, “bodies possess inertia,†is also dubious. Inertia, as the term is used in physics, isn’t quantifiable. Objects only have inertia relative to other objects.
Finally, I don’t think there really is a “basic philosophical problem in physics.†Matter is moved by applying a force.
I think the way you tie inertia to the basic concept of forces is very sensible. I wonder how this would translate into General Relativity where you have the curvature of spacetime rather than the gravitational force?
I am not sure I really understand your original uncomfortableness concerning inertia. Why is it puzzling as to what you are working against when you work against inertia? Although I can say that the idea that inertia is related to “the fabric of the cosmos” or the universe of distant masses is a philosophical argument many physicists have found appealing that goes by the name of Mach’s Principle. No one has turned this principle into a well established theory, but many have tried.
The basic questions of physics are inherently about the world we perceive and not so much about the “things-in-themselves”. Perhaps we shouldn’t be staying up all night worrying about them, but they are no less knowable then any other fundamental question.
Movement is a relationship of phenomenon to observing phenomenon - a comparison.
Therefore inertia signifies the absence of a relationship and so alludes an absolute, which is non-existence, itself.
Existence, as I define it, is what has a temporal character, that is it moves and acts and changes in relation to what is observing it or conscious of it.
Life, consciousness itself, is activity, temporality, that has ordered itself to such an extent - attained some level of absoluteness - that is can focus its energies upon specific probabilities.
Movement is a disparity of temporal flow.
Temporality exhibits this pooling disparity because it is unstable and deteriorating into entropy - which means that it is fragmenting into more and more unities and so relationships creating an inability to perceive patterns by a mind like ours, which we call chaos.
This instability is a lack of an absolute, which a conscious mind (an emerging temporal unity) interprets as need/suffering.
So inertia is a concept insinuating the absent absolute, since to imagine inertia we imagine the stable, unchanging, unmoving and so timeless.
It is non-existence as imagined by the existent.
Asok: (Sorry for the delay - I’ve only just seen the replies!) What I meant by the velocity statement is that there’s an absolute sense in which things can move with a constant velocity (inc, of course, zero); that acceleration can always be detected/felt.
Re what I called the basic philosophical problem in physics, I think you’re deferring it, not answering it. If I’ve got two atoms in empty space, with no other atoms for light years around, why do they move towards each other? We can say how they’ll move, but - fundamentally - not why.
Seb: I’m glad you sympathise with my views 
Re general relativity, I’m not sure about that, as I’ve never studied it (only did special). Re Mach’s principle, yes, this is a tempting explanation, but even if it’s true there’s still the problem of what animates matter in the first place.
Satyr: I don’t follow what you’re saying, although parts of it look good.