Two issues have interested me for a long time. One is Kant's perception of metaphysics as an illusion-prone area, while the other involves the intriguing way mathematicians expand their concepts. Although mathematicians may talk about the sine of a complex number, they do not try to define the sine function to apply to the moon. The connection between the two areas becomes clearer when one recalls that Kant has argued that the antinomies of reason derive from illegitimate expansions of concepts beyond their range of application (e.g., applying the categories of causality to the whole world of phenomena). It is important to note that this connection has reappeared and even intensified in contemporary thought. One cannot imagine modern mathematics and physics without the procedure of expansions of concepts, and the analyses of Russell's paradoxes by Russell, Gödel, and others are echoes of Kant's view that one cannot view certain totalities as genuine objects. A parallel approach, with certain changes, may even be attributed to Wittgenstein, whose aim in philosophy was to “bring words back home,” as well as Brouwer's diagnosis that classical logic was derived from a careless expansion of logical laws that are valid for finite collections to unbounded ranges.

These developments raise an interesting question about Kant's analysis of the source of illusions, as they make it clear that modern scientists and mathematicians do not respect the boundaries within which the concepts they use were originally defined.