I'm tempted to call upon F = MA and define it in terms of mass and acceleration. Except that we would need a definition of mass, and mass is often defined as a measure of inertia, of how much a given force will accelerate an object. And that threatens to be circular.
F=ma is not circular. However, it requires that two of the three quantities be defined independently. Having done that, the third quantity is then
defined by the equation.
The definition of acceleration is relatively simple and transparent. With that in place, we then have a choice: we can either define mass and let F=ma define what force is, or we can define force and let F=ma define what mass is.
The original approach (back in Newton's time) essentially chose to define force in a common-sense way, as the strength of a push or a pull. In practice, a reliably reproducible force could be produced experimentally in various ways. One way would be to use a spring, whose force varies linearly with the distance it is compressed or stretched. Experimentally, then, we find that the measured acceleration of a mass pushed by a compressed spring (say) turns out to be directly proportional to the applied force.* If, for example, we draw a graph of F vs a, then we find a straight-line relationship, experimentally. We then
define the gradient of that straight line to be the mass of the pushed object. Intuitively, mass is a measure of the object's inertia, or resistance to acceleration.
The other approach - the more modern one - is to start by defining mass. Mass, roughly speaking, is currently defined as a certain measure of the amount of "stuff" in an object. Regular macroscopic objects are made of atoms, each of which has a characteristic mass. The mass of an object is, in principle, just the sum of the masses of its constituent atoms. Having defined mass and acceleration, we then
define force F as the force required to give an object of mass m an acceleration a, using F=ma.
There are more details, of course. Until around 2019 (?) the standard kilogram was defined with reference to a specific physical object kept in Paris. Now it is defined with reference to Planck's constant, which itself now has a defined value in the SI unit system. The reason for doing it this way is that Planck's constant eventually became measurable to far greater accuracy than the mass of the standard Paris kilogram cylinder.
I guess that sometimes I think of ideas like 'force' and 'energy' when they are being used in physics, as something akin to accounting gimmicks, quantities that make the equations of physics come out as desired when measurements of quantities that are more directly observable are plugged into the equations.
That's true to an extent. However, definitions like F=ma are not completely arbitrary. They are defined the way they are because they make the prediction and interpretation of experimental results easier. Suppose that instead of defining force F by F=ma, we chose instead to define force as F=ma^2. On paper, there's no problem with that (other than raising calculational and theoretical difficulties), but experimentally it vastly complicates things. Now, force (as defined) has a non-linear relationship to acceleration. Double the force on a given mass and the acceleration now is found experimentally to increase by a factor of sqrt(2). In simple cases, such things might not matter much, but a lot of
other physical principles depend on the definition of force. Over all, it makes for a massively greater amount of work and complexity to start with F=ma^2 rather than the more intuitive F=ma.
Occam's razor suggests that scientific theories should strive to use the simplest possible theory that explains phenomena. If we put F=ma up against F=ma^2, then the former definition wins the battle for simplicity hands down.
I might also mention that, of course, the intuitive notion we have of a force as a push or a pull is more in line with common sense idea of F=ma, as opposed to F=ma^2. We expect that if we push something twice as hard (according to our perception), then something about its motion should be twice as big, not sqrt(2) times as big. Of course, it took us centuries to appreciate what the particular "something" that doubles actually was, so it's not all about "common sense".
Action at a distance is an age-old question and there's a huge literature on it. It was a huge topic of discussion and controversy in classical physics and despite assertions that it's been solved today, I'm still not convinced. The argument often seems to be something along the lines that early modern atomistic 'billiard-ball physics' only recognized transmission of force by physical collisions. But we don't think that way any longer, so it's no longer an issue for us. Which kind of reduces it to it just happens, accept it.
Or that it's implicit in the idea of a field, which threatens to become circular if fields are defined in terms of action at a distance. Just giving something a name doesn't supply it with a mechanism.
A physical field, fundamentally, is something that has a value at every point in space at all times (different values at different points). The modern picture of interactions is that "forces" are transmitted by excitations (a bit like waves, but also a bit like particles) in the relevant fields. This is not, technically, action at a distance, because something has to physically travel from the "source" location to the point where the "force" is felt. What travels are those carrier bosons you mentioned.
I guess that Einstein tried to explain it for gravity in terms of the geometry of the space in which an object is located, but I don't know how generalizable that approach is to other actions at a distance.
This is where I start to get out of my depth. However, I think that string theories are attempts to generalise that approach beyond gravity. This seems to be possible, but it comes at the expense of introducing extra dimensions of space which are currently unmeasurable.
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* Perhaps using a spring is not the best example here, because the spring force varies continuously as compressed a spring pushes on an object. But you get the idea, hopefully.
P.S. Thinking about this some more, we could use a compressed spring to push on and accelerate an object with a constant force. We just have to make sure that we move the spring with the object in such a way that the compression of the spring stays constant during the object's acceleration.