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Kant was very proud of his Copernican revolution. So it is a bit ironical that the exact nature of this revolution should have turned out to be as obscure and controversial as it has. In the present paper I will try to provide a new way of looking at the issue. It is my hope that this new perspective will prove not only historically but also theoretically valuable; in particular, that it will present Kant's revolution as one that we might want to take seriously, and maybe even think we still need.
Wittgenstein's thought on mathematics had undergone a major, if often undetected, change. The idea that adopting an algorithm like "plus" determines in some physical, mental, or metaphysical way one's response to infinitely many exercises is nothing but covert Platonism, in many ways worse than the Platonism of objects. Wittgenstein agrees entirely with the Intuitionist critique of the law of excluded middle. For the Goldbach conjecture to be true in the sense of classical mathematics, we have to say that the operations of arithmetic determine in advance that every even number, no matter how large, can be partitioned into two primes. The law of excluded middle cannot be regarded as a hardened regularity in cases in which it is applied it to a putative infinite totality. But precisely because of this, there is no direct comparison possible between empirical observations and mathematical theorems in this type of proof.
In his lucid and perceptive essay, “Recent Work on Kant's Theoretical Philosophy”, Karl Ameriks signals Kant's distinction between appearances and things in themselves as one of the (two) “central issues” of the Critique of Pure Reason. The reason why the issue is central (and complicated) is that Kant appears to say contradictory things on the matter. At times he says (or implies) that appearances are the same as things in themselves, and at other times he says (or implies) that they are different. Some interpreters have tried to make sense of these contradictions by claiming that “although for Kant there are not two objects involved, there are still two transcendental and intelligible aspects or points of view that are called for by his doctrine of things in themselves and appearances”. However, it is not immediately clear what kind of an animal an aspect or a point of view is, what kind of operation it is to “look at” an object from such different points of view, and what kind of results this operation is supposed to give. In the present paper, I make a fresh proposal. I propose to interpret Kant's conflicting claims on the relation between things in themselves and appearances in terms of the contemporary framework of possible-world semantics.