Gary J. Richmond

Questions regarding the nature and acquisition of mathematical knowledge are perhaps as old as mathematical thinking itself, while fundamental issues of mathematical ontology and epistemology have direct bearing on mathematical cognition. Several original contributions to logic and mathematics made by the American polymath, Charles Sanders Peirce, are of direct relevance to these fundamental issues. This chapter explores scientific reasoning as it relates to abduction, a name that Peirce coined for educated “guessing” of hypotheses, which he took to be “the first step of scientific reasoning” and the only creative one. Yet he also argued that all deductive reasoning is mathematical and that all mathematical reasoning is diagrammatic. Representation, especially in the form of a diagrammatic system of logic that Peirce developed, is explored here along with his logic of inquiry, most notably in terms of its manifestation as the logic of ingenuity. Originating in the field of engineering, here the diagram of a problem serves as a heuristic substitute for evaluating the actual situation, an approach that can be extended to other forms of practical reasoning such as ethical deliberation. This chapter also touches upon such diverse but related subjects as non-Euclidean geometry and nonclassical logic, with additional examples that help to elucidate cognitive elements of mathematical knowledge.