I wrote this brilliant answer this morning, and then I hit the wrong key on my crackberry and it's gone forever!
Part of our problem is the absence of shared vocabulary and context.
I mentioned the prime mover as an example of something immaterial having material effects. If God can do it, he could enable the angels to do it.
In the modern geometry in learned in high school “point” was undefined. Euclid (Heath translation) says “A point is that which has no part.”
His second definition, which is charming, is, “A line is a breadthless length.”
Now I THINK both Euclid and say, Lobachevsky, thought they were writing about real things. But, of course, the term “real” is loaded, and doesn't necessarily mean “material” or even “empirically observable.”
I don't know how to “prove ...that a point occupies no space.” It's what we mean by point, I would have said. When you think of Latitude 0 deg 0 minutes 0 seconds and Longitude 0 deg 0 minutes 0 seconds, are you thinking of an area?
“...and any linear expression of space is composed of points along the linear span. “ At least back when the world was young there was an infinite multitude of points in any linear expression of space. If we sort of do lightning math history for dilettantes and we have the Cartesian line (1,0 - 2,0) Dedekind will assert (and appear to prove — to my satisfaction) that between any two given points there is another point. And if that is true then the operation of denominating that point can proceed without end. and if THAT’s right, then there is an infinite multitude of points in a linear span. Yes?
Mind you, In a Philosophical Theology for Amateurs 101 class I taught for a few weeks mumble mumble years ago, I drew a round thing on the blackboard and asked what it was. A kid said, “A circle.” I said, “No, it's a PICTURE of a circle.”
So that's how I think, Gawd ‘elp me.
In other words, as Quix knows (and places his face firmly in his palm in my direction when he remembers it), I think there are no examples of a triangle (or of triangularity) and at most only one enduring and perfect (so far) example of “human nature”. But Triangularity and Human Nature are “real”. Not only that, but they are, in a way, realities whose existence we deduce from observation. But what we observe approximates the “real” thing.
(Excuse me while I go remind myself what the heck Planck's constant is ....) Okay, the size of a “so much” (quantum) or the amount of energy or, wait, my head hurts.
Now, what I am doomed by my fate to wrestle with is, “is it meaningful that I can express Planck's Constant, slap parentheses around it, draw a line under it and a 2 under the line? I'll get a meaningful mathematical expression — meaningful in the sense that it doesn't violate the rules of mathematics or lead to an absurdity. Just 'cause I won't see something that size, doesn't mean (to me) that it's not a ‘real’ value. It's just not one I'm likely to run across a whole lot, if ever.
Mutatis mutandis for 1.616252(81)×10−35 metres. If I can say that, then I can say [1.616252(81)×10−35]/2 metres, and [1.616252(81)×10−35]/2 is a real number.
(Let's all pause and give thanks for Wikipedia.)
I guess to bring it all back home, IF one acknowledges the "reality" of, say, "circularity", then one acknowledges a kind of reality which is not "material", "natural", or "physical".
AND, stuff in the "material", "natural", or "physical" sort of reality seems "ordered to" this immaterial reality. Things are ALMOST circular or triangular. Deeds are more or less "just", objects are more or less "beautiful."
And once you've done that, the cat is out of the bag. We can have entities, beings, which are utterly immaterial and which still influence material stuff.
And so, as many angels as you please (and even more!) can dance on the rough surface of the salt-rimmed Margarita glass. And the ones who fall in will consider themselves especially blessed.