This paper is intended as a contribution to the longstanding debate about how to understand Kant's appeal to pure intuition in his philosophical account of geometry. It proceeds by focusing on a notion that has been at the center of this debate, namely, the "construction" or "exhibition" of a concept in intuition. I claim that not only does Kant appeal to this notion in two different ways—in support of both an epistemological and a semantic thesis, each based in his broader philosophical project—but also that the demands placed on this notion by these appeals are in conflict. The tension between the epistemological and semantic appeals to construction-in-intuition can be clearly seen in geometrical proofs that proceed by reductio ad absurdum. I conclude by considering the prospects for resolving this tension in a manner consistent with Kant's broader philosophical ambitions.
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