Point taken. Still, if you don’t mind me trying to dot the i’s and cross the t’s (or was it the other way around?) space in the general sense cannot be a Platonic ideal since that is an abstraction of concrete existing objects (actually Plato would of course say it’s the other way around). Space in the sense of the space taken by a particular object might be a Platonic ideal, but in fact falls under the same criticism. (It is true that you didn’t claim either of this: your point is that space has similar features to a Platonic ideal.)
Space in the general sense can be a Kantian a priori concept.
Gosh, methinks we’ve come to a satisfactory conclusion.
I think so. I can live with a synthetic a priori claim about space. I probably wouldn’t go so far as Kant may have, but that is with the hindsight of all kinds of geometries not available to his mind (which, I think, may be a great shame, I have no doubt that his metaphysics were stunning, but his morality troubles me). Still, though Hume inspired Kant I find myself much more on Hume’s side for most metaphysical subjects he tackled (I must admit I’ve read very little moral philosophy at all).
Again, though, space in the sense of “the space taken by a particular object” is not quite what I’m aiming at. Rather, I’m aiming at the space around an object that we may move it into, or back into again. This space, as I’ve attempted to describe it, has the hallmarks of transcendental realism: an essence or abstraction hinted at by the senses but available only to the intellect; and, in knowing it through the intellect, revealing truths about the nature of, well, stuff. It is not to posit two entities where there is one, it is to posit one transcendent universal (space<->shape) where there are many “imperfect” realizations/manifestations (actual objects). The immanent property of this apple’s shape is fully realized by the ideal “shape”—that is, by its relation to space, from which it is derived. It also seems to have the same circulararity as I’ve found of most transcendental ideals, where we come to understand the ideal apple by thinking on the apple in front of me, and once we understand the ideal we use it to discuss the apple in front of me.
With respect to a link between shape and space, I’m still not sure whether I can fully agree with you. When speaking about shape or space we generally mean different things, even though they have in common that we cannot see them directly. The space taken by the object has a specific place, while the shape of the object is moved along with the object.
Furthermore I do not feel that in talking about shape or space we are already talking of an ideal in the usual sense of the word. We talk about something that can only be grasped intellectually, true, but it is still not ideal in the sense of abstracted from its accidental properties.
As a final note, I do not feel that ideals are actually defined in a circular manner. The easiest example is again something like the triangle. Most (all?) actually drawn triangles are not perfect triangles, especially those drawn loosely on a blackboard. Still no-one has problems using such a drawing to make the intellectual leap to the idea of a triangle. Then, when actually applying mathematics (for example in building or civil engineering) such ideal triangles etc. may be used as a guideline for setting up an actual building. You may call this circular if you will, but it is not a simple or vicious circle.
(Starts singing “I feel nit-picky” 
)
Of course. Just like when we’re talking about instantiated white we’re not talking about what the universal white is like. Scope is not a factor in the equation I’m trying to make
I don’t know why you say this. What accidental properties remain?
I meant that the shape/space of an actual badly-drawn triangle is bumpy and curved, while the idea taken out of that triangle (to my mind, at least) has perfectly straight sides. When writing that sentence in my last post I was already thinking that it might prompt such a question; I should have reconsidered. It is true that (to use my example again) the shape of a non-ideal drawn triangle does lose some properties in comparison to the real object, such as it being drawn in chalk.
(to answer the objection that a triangle arguably is a two-dimensional object that does not occupy space, please change all the above to models of a pyramid, or a football if you so please)