On Hilbert and Kant: i) Hilbert's quotation looks as if it might be by Kant as well; the style is certainly fairly Kantian. ii) I can't locate it in the Critique of Pure Reason, or in other works by Kant, but I haven't looked very hard. iii) I think it's clear why Hilbert would have preferred the version he had: an Idee, for Kant, is a concept that is produced by reason (Vernunft), which transcends the bounds of possible experience, but which has a regulative role in thought. This is very like Hilbert's concept of what he called ideal mathematical assertions.
(Incidentally, all of the terms here -- Anschauung, Begriff, Vernunft, even Sinnen -- are technical terms for Kant, so you can't really figure out their meaning from their meanings in contemporary German.)
I have a question, which is this: does Hilbert give the reference for the Kant citation, or was it inserted by an editor?
Graham White Department of Computer Science Queen Mary and Westfield College