Kant's schematism and the foundations of mathematics

Abstract

The theory of schematism was initiated by I. Kant, who, however, was never precise with respect to what he understood under this theory. I give—based on the theoretical works of Kant—an interpretation of the most important aspects of Kant’s theory of schematism. In doing this I show how schematism can form a point of departure for a reinterpretation of Kant’s theory of knowledge. This can be done by letting the concept of schema be the central concept. I show how strange passages in, say, the first Critique are in fact understandable, when one takes schematism serious. Likewise, I show how we—on the background of schematism—get a characterization of Kant’s concept of ‘object’. This takes me to an analysis of the ontology and epistemology of mathematics. Kant understood himself as a philosopher in contact with science. It was science which he wanted to provide a foundation for. I show that, contrary to Kant’s own intentions, he was not up-to-date on mathematics. And in fact, it was because of this that it was possible for him to formulate his rather rigid theory concerning the unique characterizations of intuition and understanding. I show how phenomena in the mathematics of the time of Kant should have had an effect on him. He should have remained more critical towards his formulation and demarcation of intuition, understanding and reason. Finally I show how D. Hilbert in fact gives the necessary generalization of Kant’s philosophy. This generalization provides us with a general frame work, which functions as a foundation for an understanding of the epistemology and ontology of mathematics.