Sorry to dismiss your question: it is actually a very good question, in the manner I interpret it.
The question of Kant and Schopenhauer is one of the most difficult in philosophy, but in many ways transcendental idealism is the only philosophy, something we should be wary of saying in public. It is almost like quantum mechanics (a totally different subject): noone really understands it. But a strong glimpse of the basics can be enlightening. Kant picked up the start made by Plato and all the rest of philosophy is mostly a wasteland. Schopenhauer tried to stream line the subject and the result is take it or leave it: no proofs at all. I am supremely fortunate in having arrived at Kantian insight (not til age 50, please note) by stepping backwards into it trying to solve Kant’s challenge. I was provided by an example of something that made transcendental idealism make sense. I am also lucky in never having taken a college course in Kant (very valuable, absolutely essential, in many cases, but …) being able to find my way to the subject via an exotic study of history.
My post I should set aside for a moment: it was close to a breakthrough linking Advaita, Schopenhauer, Bennett, Samkhya and the free will/ethics question. but I got a bit sloppy and made some statements I am sure Schopenhauer wouldn’t accept. To locate the will in the noumenal is that breakthrough, but it is not in Schopenhauer in quite that way, and requires a more careful thinking through. However, a version of that is clearly in Schopenhauer. The point is that if the source of action/decision, choice is in the noumenal its causal outcome is not a contradiction to free will claims. that’s the simple and basic set of issues (getting it straight via Kant/Schopenhauer can result in multiple drafts slightly off like my post) that students of physics are simply incapable of grasping. That doesn’t mean we are out of the woods. I was troubleshooting the whole terrain between Schopenhauer, and Advaita, and I will solve the whole question soon, I hope. Every new ager should say hallelujah for Schopenhauer: he found the way to a
The issue of proof you raise is quite OK. btw, Kant had an exotic proof discourse related to the synthetic a priori, very controversial. Challenge that, and many have, and Kant has no proofs of anything. But his basic point comes via that other area of his thought, the Transcendental Deduction. Your point on proof assuming I understand what you are saying is thus understandable, but Kant’s point is that space and time are among the categories of perception and this suggests something amazing: that space/time are constructs of the mind. the issue is highly complicated and never appears in Schopenhauer. Here Kant has the whole of philosophy by the throat, small wonder people limp away, no proofs, no chop suey. If Kant is right then…
there is a better way here: pick up a copy of bryan magee’s Confessions of a Philosopher: he leads the reader through baby steps into the whole subject without any of the hypercomplexities. Just using the antinomies in the Dialectic, the last section of the Critique of Pure Reason, shows something the earlier chapters often fail to do. These antinomies are the teenager’s golden path to Kant 101.
There are many versions, one in my Enigma of the Axial Age, from Plato: the edge of space antinomy: if I reach the edge of space, can I reach beyond it? Another, there is a beginning in time, no there is no beginning in time. these two examples show that Kant is onto something: space and time are contradictory in our experience. physicists start gasping here, but the demonstration is clear, and so simple that people used to advanced physics can’t conceive it could be that simple.
The antinomies show that our experience is contradictory and can’t be accounted for on the basis of simple realism. That is the road to transcendental idealism.
There is more here, and I hope I will be able to sort out the issues a little better here, and especially it would be nice to clarify Advaita, the one subject that is arguably more advanced than Kant, but its format, like that of samkhya, has been scrambled.
More on this later, but we are close to an immense unification and I think that will include the buddhist canon which is finally a family member in the Indic legacy.
I would not dismiss Schopenhauer too cavalierly. He is the key to a new and future modernity that can handle the buddhism you revere.