Imagination, Concentration, and Generalization: Peirce on the Reasoning Abilities of the Mathematician

I discuss the epistemic conditions for the possibility of mathematical discovery that are implied by Peirce’s logic of mathematical inquiry. Peirce describes the mathematician’s reasoning abilities as the powers of imagination, concentration, and generalization. I interpret all three as different semiotic abilities to reason with mathematical icons, given Peirce’s conception of mathematics as the study of what is true of hypothetical states of things and his view of mathematical method as experimentation upon diagrams that embody formal relations. These abilities come into play at different stages of the process of experimentation upon diagrams. I illustrate Peirce’s view with a historical example from ancient Greek geometry.