I used to get confused when I first learned the electromagnetic field is said to have a momentum. A few years later, I came across an argument for why 0.999... = 1: When faced with the task of defining what a recurring decimal means, we as a community decided to make the definition consistent with the rules of algebra of "normal" numbers, paying the cost of this prima facie counter-intuitive result.
When faced with a similar task in electrodynamics, we apparently chose to keep the theory consistent with the conservation laws of mechanics, at the cost of assigning energy and momentum to seemingly "non-corporeal" entities called fields.
Why was this choice made? Why salvage conservation laws that seem "wrong"?