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To distinguish the word "arbitrary" from "random", I think of an arbitrator—i.e., an outside judge chooses something. (Maybe this results in a uniform prior for me, if'n I don't know what she'll do. Or maybe I'm a mathematician and I choose to be ready for any choice that arbitrator might make.)

When I'm teaching linear algebra and explain arbitrary parameters to my students, I use exactly this metaphor. How many times does someone else have to come in and arbitrate the value of other variables, before you can tell the questioner what the answer is?

I agree with those who say it's okay to figure things out later. If my music professor says a certain composer favors the Aeolian mode, I may not be able to visualize that on the spot but who cares? I can remember that statement and think about it later. Likewise with phlogiston, I have a vague concept of what it is and someday the alchemists will discover more precisely what's going on there.

Too much cognitive effort would be spent if, every time I thought about linear algebra, I had to visualize the myriad concrete instances in which it will be applied. I bet thinking in abstractions results in way more economical use of thinking time and thinking-matter.