If we can construe a concept with an object (via a schema), then we know that it is at least the concept of an ens imaginarium (not, like the concept diangle, that of a nihil negativum [A291/B348]; and not, of course, of an ens rationis or nihil privativum, either).
But I now think I was misunderstanding the relationship between ens imaginarium and nihil negativum. A nihil negativum is indeed, in a sense, less (more nothing) than is an ens imaginarium, but only because the concept of it is more fully objective. That is: an ens imaginarium cannot properly be called either objectively, that is, transcendentally possible or impossible. (It must be subjectively, that is, formally possible; even an ens rationis is that much.) An ens imaginarium is without substance (A291/B347), i.e. without relation, and so the issue of possibility, i.e. modality, does not arise.
This must then affect the suggested interpretation of konstruieren. Rather than saying that the Konstruktion promotes a concept to the level where it becomes the concept of an ens imaginarium, but that then some further step is needed to promote it to the concept of a possible being, I should have said that the concept of an ens imaginarium is a concept which lacks a Konstruktion. Such a concept is not fully objective in that it fails to posit its object as heterogeneous to a ground of possibility. In particular, according to the thesis of transcendental idealism: such a concept fails to posit its object as heterogeneous to consciousness (pure space and time, entia imaginaria, are in us). A construed concept (konstruierte Begriff, see A716/B744), on the other hand, is a concept taken as a (purported) determination of the general (transcendental) relation between consciousness and its external object. As such it will be the concept of a possible object if the general relation admits of being determined in that way and, if not, not.
Our intellect is discursive. That is: there is for us a pure form of intuition, which puts specific conditions, not derivable from the unity of apperception itself, on the manifold which is to be unified in our having an object. (We have a particular species, Art, of intuition.) It is thanks to such conditions that the general relation in question, in our case i.e. the formal conditions of experience (again: A218/B265) admit of some determinations (e.g., determination to perception of a triangle) and not others (e.g., determination to perception of a diangle).1 And so, finally, what Kant actually says about the diangle (that is: about a non-Euclidean figure) in the proof of the First Postulate is that its impossibility
rests not on the concept [of the diangle] itself as such [an sich selbst], but rather on [the concepts]2 Konstruktion in space, i.e. on the conditions of space and the determination of it [desselben, = of space]; these, in turn, have their objective reality, i.e., they pertain to possible things, because they include in themselves the form of experience in general. (KdrV A221/B268)
Which I take to mean: the geometrical Konstruktion of the concept diangle is a construal in the above sense, that is a construal upon which the possibility or, in this case, impossibility of the concepts object can rest, because the conditions of space are (among) the specific conditions of our experience in general.
Then this is the same as what he says a little later about the concept triangle (the passage I already quote in my previous post): that its object would remain always only a product of the imagination (i.e., an ens imaginarium) if it were not that
The very same [eben dieselbe] imaging synthesis by which we construe a triangle [with its concept] in the imagination is wholly the same as [mit einerlei] that which we exercise in the apprehension of an appearance, in order to make from it an empirical concept. (A224/B271)
That is: the procedure of the imagination in schematizing the concept makes it more than just a product of the imagination (posits it as heterogeneous to the imagination itself) only because the imagination in that procedure is subject to the same conditions to which it would be subject in construing an empirical concept with an object: that is, in experiencing such an object.3 The latter conditions the conditions on experiencing the object of an empirical concept are the formal conditions of experience in general.
This apparatus is what allows Kant to say that the Euclideanness of space is a special case, while at the same time denying that we know the possibility of any other case. We know the formal possibility of the concepts (such as diangle) which might function is such other cases, but we dont know the possibility of such alternate geometries as geometries. That is: we dont know the possibility of alternate specific formal conditions of experience in general. Because pure space and time are themselves mere entia imaginaria, we dont even know the possibility of our own geometry except insofar as it is actual.
Im interested in all of this, in part, because of a question I asked in a previous post, namely: what are the objects of mathematics? (That is: what should we say they are, taking that Kant must be at least partly wrong about that.) I want to suggest something like: that an object of mathematics is a kind of deconstrual of a concept, or of a type of concept. So, for example, one object of mathematics would be: the deconstrual of geometrical concepts which displays Euclidean geometry as a special case. Here deconstrual must be a form of determinate negation. What was the construal becomes absorbed into the formal content of the concept itself, and so abstraction from it becomes possible. (But, Im suggesting, its the deconstrual itself, rather than the abstraction it subsequently allows, which is an object of mathematics.)
1In an intuitive intellect, so to speak, every representation which is not itself formally deficient is the representation of something possible. There is no other form with which it must be compared. But so to speak already displays how little we really understand the possibility of an intuitive intellect. I mean: that we do not know what are the formal conditions on such representations (their conditions qua determinations of the consciousness which is their subject). The law of noncontradiction is a formal restriction on discursive intellectual representations, i.e. universal concepts.
2This is the meaning according to the text of A and B. The 5th edition, apparently, reads derselben instead of desselben, so according to that we would have to understand [the figures] Konstruktion. As I pointed out in my previous post,
Kant speaks in both ways.
3Kant often, but I think not always, restricts Konstruktion to (what I am calling) the construal of a concept in the case where this can be done a priori (including, apparently, in the passage Ive been discussing from A2234/B271). I take it it is not too great a stretch to extend the term to the case of empirical concepts, simply by removing the qualification a priori in the definition ihm gänzlich a priori einen Gegenstand geben (A223/B271) or die ihm korrespondierende Anschauung a priori darstellen (A713/B741).