Kant’s opening moves in setting up a metaphysics that will refute classical metaphysics are to claim that human knowledge is limited to intuited objects, and that we intuit only sense objects. If true, the refutation is a fait accompli, since classical metaphysics is of objects for which there is no sense intuition.*
Let’s take it as axiomatic that sense intuition is primary. Kant takes this primacy as a limit that makes everything beyond it unknowable or even meaningless, and the Nietzschean/Empiricist/Naturalist tradition that came after Kant followed him in this. The classical tradition has two different approaches: in the Platonic tradition sense intuition is of a reality that takes part in an object that we intuit by intelligence; in the Aristotelian tradition there is no intuition outside of sense but we can judge things intuited by sense to be the appearances of things unseen.
There are two reasons to think that sense intuition cannot constitute a limit on knowledge: (a) Any limit to knowledge is contextualized by the known and (b) sense intuitions are essentially passive while knowledge is essentially active. For (a) while it is possible for a physical limit to not be contextualized by something physical (the boundaries of the universe, say) cognitive limits can only be contextualized by something noetic. Vision, for example, is limited by darkness and hyper-luminosity, both of which are seen; and silence is detected with the same organs as sounds. In the intelligible order what exists is contextualized by our awareness of its contradictory through the principle of contradiction. In other words, the noetic order differs from the real by the fact that opposites always exist for it. For (b) knowledge is not a way of suffering something but of perfecting a cognitive being interiorly. We don’t explain our knowledge by speaking of how we receive the world but of how the world, precisely as other, is our own.
What we say here will also be applicable to the (often poorly formulated) question of the existence of mathematical and logical things.