In The Critique of Pure Reason, Kant distinguished between the phenomenal world (the world as representation) and the noumenal world (the world as it is in itself). The former refers to the world as we experience it; the latter refers to the world as it exists independently of our experience. My question concerns whether Kant is justified in positing the existence of the noumenal world. I don't think that Kant's project can properly be appreciated without first considering the context within which he is writing, specifically, the sceptical challenge laid down by Hume and the advances in the natural sciences reached by Newton. As a caveat, I have not studied this topic formally but have instead relied on self-study and, as such, I apologise in advance if my explication of said topic is superficial.

Hume distinguished between relations of ideas and matters of fact, hence the term 'Hume's fork'. Hume's fork can be stated thus:

(1) Statements about ideas. These are analytic, necessary, and knowable a priori.
(2) Statements about the world. These are synthetic, contingent, and knowable a posteriori.

In Kantian terminology, members of (1) are known as analytic propositions and members of (2) are known as synthetic propositions.

Herein lies the crux of Hume's sceptical challenge. Hume points out that synthetic statements are not certain since it is logically possible that any given statement about the world is false. Moreover, analytic statements can only be used to prove other analytic statements, and mean nothing outside of the context of how they relate to each other, and therefore tell us nothing about the world.

In effect, Hume is stating that analytic statements are certain, but they tell us nothing about the world. Since they tell us nothing about the world, they cannot be used to prove synthetic statements. Hence, synthetic statements have no certainty. This meant that the Newtonian physics of Hume's day was uncertain. Now we turn to Kant.

Kant agrees with Hume that all knowledge begins with experience. However, Kant argues that it does not follow from this that all knowledge arises from experience. To quote Kenny:

[Kant] seeks to show that without the metaphysical concepts that Hume sought to dismantle, Hume's own basic items of experience, impressions, and ideas would themselves disintegrate

(A New History of Western Philosophy)

Kant argues that Hume overlooked the existence of synthetic a priori statements. All metaphysical statements, Kant maintains, are of such a kind. We apply metaphysical concepts to the world in order to make experience possible at all. In other words, synthetic a priori metaphysical concepts are the pre-conditions of all experience. Hence, Kant distinguishes between the world as experience it (the world as it is experienced given the application of said concepts) and the world as it is independent of our experience.

It seems to me that Kant presupposes that there exists a world which, by virtue of its being independent of our experience, is unknowable, yet nevertheless is the cause of our experience. This presupposition seems to me unjustified. How does Kant know that such a world exists?

To quote Nietzsche in Twilight of the Idols:

The real world, unattainable, undemonstrable...And if unattained also unknown...The 'real world' - an idea no longer of any use...an idea grown useless, superfluous, consequently a refuted idea: let us abolish it!...We have abolished the real world: what world is left? the apparent world perhaps?...But no! with the real world we have also abolished the apparent world!'

The problem, it seems to me, is that if we 'abolish' the noumenal/real world then our a priori metaphysical concepts are not applied to anything. Perhaps I am presupposing the validity of subject-predicate logic here? i.e. If there exists a predicate then there must exist a subject to which that predicate belongs; if there exist a priori metaphysical concepts then there must exist a world to which they are applied(?)

Thanks in advance.