The notion of the "construction" or "exhibition" of a concept in intuition is central to Kant's philosophical account of geometry. Kant invokes this notion in all of his major Critical Era discussions of mathematics. The most extended discussion of mathematics, and geometry more specifically, occurs in "The Discipline of Pure Reason in its Dogmatic Employment." In this later section of the Critique, Kant makes it clear that construction-in-intuition is central to his philosophy of mathematics by presenting it as the defining (...) characteristic of mathematical knowledge. He claims: [M]athematical knowledge is the knowledge gained by reason from the construction of concepts. To construct a concept means to exhibit a... (shrink)

Kant repeatedly uses the biangle as an example of an impossible figure. In this paper, I offer an account of these passages and their significance for the possibility of geometry as a science. According to Kant, the constructibility of the biangle would signal the failure of geometry. Whereas Wolff derives the no-biangle proposition from the axiom that between two points there can be only one straight line, Kant gives it axiomatic status as a synthetic a priori principle possessing immediate certainty. (...) Because we are unable to generate a schema for the biangle, the failure of the attempt to construct it is intuitively clear. The parallel between mathematical and empirical concepts is instructive because both involve the synthesis of disparate intuitions into a unity. We do not, strictly speaking, even possess a well-formed concept of the biangle, because its representation cannot fulfill certain basic requirements of concept formation. (shrink)