Kant’s denial that psychology is a properly so-called natural science, owing to the lack of application of mathematics to inner sense, has garnered a great deal of attention from scholars. Although the interpretations of this claim are diverse, commentators by and large fail to ground their views on an account of Kant’s conception of applied mathematics. In this article, I develop such an account, according to which the application of mathematics to a natural science requires both a mathematical representation and (...) a metaphysical validation for the positive use of this representation to achieve a priori knowledge about nature. The second condition—that of metaphysical validation—has been overlooked in the literature. I show that psychology’s falling short of natural scientific propriety consists not in our lacking sufficient mathematical tools for the representation of inner states, according to Kant. After all, we can represent the mere temporality of inner states with the line and their intensities with numbers. Rather, the problem is that metaphysics does not validate the further use of such mathematical entities for the achievement of a priori knowledge about inner phenomena. (shrink)

Descartes’s most extensive discussion of the law of refraction and the shape of the anaclastic lens is contained in Rule 8 of "Rules for the Direction of the Mind". Few reconstructions of Descartes’s discovery of the law of refraction take Rule 8 as their basis. In Rule 8, Descartes denies that the law of refraction can be discovered by purely mathematical means, and he requires that the law of refraction be deduced from physical principles about natural power or force, the (...) nature of the action of light, and the behavior of light rays in a variety of transparent media. For over a century, however, there has been broad agreement that Descartes discovered the law of refraction by purely mathematical means, and that he only later provided the relevant physical rationale (via comparisons or analogies) in Dioptrics II. I execute each step in Descartes’s proposed deduction of the law of refraction and the shape of the anaclastic lens in Rule 8 and concretely show how Descartes could have discovered the law of refraction and the shape of the anaclastic by its means. Rule 8, I argue, reflects Descartes’s actual path to the discovery of the law of refraction and the shape of the anaclastic lens. (shrink)