Is the magnetic field a mathematical abstraction?
Is the magnetic field purely a mathematical abstraction or is there actually something there? In other words, if a proton floating in deep space, is there actually something that shoots out in all directions from the proton that will interact with another charge particle?

Re: Is the magnetic field a mathematical abstraction?
That is a question of philosophy.
You can measure the magnetic field, so I would say it is there. The proton does not "shoot" with anything, it just has a magnetic field around it, which is as real as the proton itself. 
Re: Is the magnetic field a mathematical abstraction?
If two protons are floating in space, they'll feel an electrical field between each other and repel, but if you see them zip by you in space at relativistic speeds, they'll behave differently from your point of view by repelling slower than they should, and you'll see a new force.
But it doesn't matter  the math was invented to describe the behavior, so ultimately it's all mathematical abstraction and you use whatever level of complexity is necessary to solve your problem. 
Re: Is the magnetic field a mathematical abstraction?
I look at it this way. If the EM field isn't real, then how can an EM wave exist? It may not be physical like a baseball, but I'd say it's as real as anything else.
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Re: Is the magnetic field a mathematical abstraction?
If you consider the relativistic effect of the motion of electrons at snail's pace, in a wire, the resulting forces of attraction and repulsion can be calculated accurately by just looking at the resulting electric forces between two wires. Or you can work it out differently (using the Lorenz  magnetic  force idea).
You can call it magnetism or not, as you please. 
Re: Is the magnetic field a mathematical abstraction?
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