For example, if we really wanted to, could we, at least in principle, model electromagnetism just considering interactions between charged particles without using the EM field? That is, is it possible to predict all the same experimental results without bringing in the EM field (or the gravitational field, in gravitational theory, etc)?
For context, one thing I remember hearing several years ago is that, in general (this was in the context of classical physics), fields are essentially just a way of preserving locality -- of explaining how a particle at one location could have an effect on a particle at a different location. It was pointed out that that we could model these interactions without fields if we really wanted to, but the math would be a lot more complicated. Is that true? If so, would the math be fundamentally different, in the sense of having to invent a whole new framework with different mathematical objects, or would the calculations just be a lot more involved?
The analogy that comes to mind is that of virtual particles vs actual particles -- if I understand correctly, the former are a handy bit of math that make certain QFT calculations much easier, but aren't actually necessary in order to explain anything, whereas the latter are needed in order to explain various observable phenomena. Are fields more like the former or the latter in that sense?
To be clear, I'm not trying to suggest that doing away with fields would be a remotely good idea, regardless of whether they're entirely necessary or not, since they're clearly useful.
Edit: Also, to be clear, I'm not asking about the mathematical definition of fields, as I already understand that. Nor am I asking about whether fields "really" exist, in the sense of corresponding to some sort of physical object, as that opens a whole can of worms about whether anything unobservable in physics "really" exists, which is an unsolved question in philosophy of science. I'm purely asking about whether fields are mathematically necessary to make all the same predictions, without inventing an entirely new mathematical framework.