The "people in times before the enlightenment just attributed everything to spirits because they didn't understand things" argument is tired and boring. Just because you're not convinced doesn't mean that it's not true, modern man.
How do you propose to experimentally verify and measure such spirits? How can we distinguish between a world in which they exist as you imagine them and a world in which they don't? How can we distinguish between a world in which they exist as you imagine them and a world in which a completely _different set of spirits following different rules, also exists. What about Djinn? Santa Claus? Demons? Fairies?
Now, do you mean measure them using our physical devices that we currently have? No, we can't do that. They are "minds beyond ours" as OP suggests, just not in the way that OP assumes.
Djinn: Demons. Santa Claus: Saint (i.e. soul of a righteous human). Demons: Demons. Fairies (real, not fairy-tale): Demons. Most spirits that you're going to run across as presenting themselves involuntarily to people are demons because demons are the ones who cause mischief. Angels don't draw attention to themselves.
An electron is a bit like a demon in the sense that you can't see one directly and we only have indirect evidence that they exist. But anyone from any culture can do an oil drop experiment and get values which aren't culture bound, at least in the long run. People have been having mystical experiences forever and the world religions still have no agreement about what they mean.