Dr. Manders is developing a novel philosophical view of mathematical knowledge that long-range intellectual accomplishment in mathematics consists primarily in making things clear by appropriate use of concepts. Dr. Manders is extending case studies testing explanations of how conceptual changes affect mathematical understanding. The studies compare how given problems are dealt with in different conceptual settings. They seek to account for historically observed differences in understandability in the different settings by theoretical models of conceptual settings and their relationships to each other. The studies concern (a) algebraic reasoning in Cartesian geometry vs. traditional figure=based geometrical reasoning; (b) the meta- mathematical notion of "model completion" as a schematic paradigm for the historical development of the number concept to include real and complex numbers, and for the movement from affine to projective viewpoint in geometry; (c) the algebraic notion of "completion of a partial group action" as a model for projective closure and conformal compactification. For example, algebraic methods promote clarity in ceratin phases of geometrical problem solving by allowing one to suppress irrelevant geometric information. Figures must by their nature display this information, whereas algebraic formulas need not. Such differences may be brought out by systematically comparing the functioning of algebraic and figure-based representation. They are expected to account for many of the historically observed differences between Cartesian and traditional geometry.
