This article addresses the learning paradox, which obliges researchers to explain how cognition can advance from a lower level of reasoning to a higher one. Although the question is at least as old as Plato, two major flaws have inhibited progress in developing solutions: the assumption that learning is an inductive process, and the conflation of epistemological and ontological questions. I adopt a radical constructivist perspective and present a few related solutions from previous mathematics education literature. I then provide a new solution that relies on Peirce's theory of abduction and Piaget's theory of operational schemes. However, with the learning paradox resolved, an ontological paradox remains: If individuals construct their mathematical realities based on their personal actions and experiences, how can we explain the predictive power of scientific hypotheses that are based on this mathematics.