https://en.wikipedia.org/wiki/Noumenon
or in other words:The difference between abstract and intuitive cognition, which Kant entirely overlooks, was the very one that ancient philosophers indicated as φαινόμενα [phainomena] and νοούμενα [nooumena]; the opposition and incommensurability between these terms proved very productive in the philosophemes of the Eleatics, in Plato's doctrine of Ideas, in the dialectic of the Megarics, and later in the scholastics, in the conflict between nominalism and realism. This latter conflict was the late development of a seed already present in the opposed tendencies of Plato and Aristotle. But Kant, who completely and irresponsibly neglected the issue for which the terms φαινομένα and νοούμενα were already in use, then took possession of the terms as if they were stray and ownerless, and used them as designations of things in themselves and their appearances. - Schopenhauer
----------------------------Fundamental error: Kant did not distinguish between the concrete, intuitive, perceptual knowledge of objects and the abstract, discursive, conceptual, knowledge of thoughts.
My claim:
If I think of a perfect circle or a perfect square, I'm having 'abstract' thoughts. It's true that 'abstract' thoughts don't refer to anything real in the noumenal external world.
If I think of the Moon or the table in front of me, I'm having 'concrete' thoughts. These thoughts are representations of real noumenal objects in the noumenal external world. (But it's also possible to think of concrete things that don't actually exist in the noumenal external world.)
So the noumenon can be used in both positive and negative sense. Which is the modern usage, as far as I'm concerned.
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VA's claim:
The noumenon can only be used in the negative sense, there can never be a real noumenal world. All noumenal references are without real referents. There is no use for an abstract/concrete distinction. Schopenhauer and many others have fallen under the very illusion that Kant has warned us about.
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So what justification could there be for taking the latter stance over the former stance?