A cursory look at the history of mathematics shows that significant progress frequently consists in relocating a set of problems in a novel conceptual setting in which they may be better understood. Based on this observation, Dr. Manders is developing a novel philosophical view of mathematical knowledge: that genuine long-range intellectual accomplishments in mathematics consists primarily in making things clear by appropriate use of concepts. Thus, the primary epistemological norms should concern clarity, and the theory must ultimately explain what it is to "make clear by appropriate use of concepts." Under this grant, Dr. Manders will extend case studies testing explanations of how conceptual changes affect mathematical understanding. The studies concern (a) algebraic reasoning in Cartesian geometry vs. traditional geometrical reasoning; and (b) the metamathematical 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. In the new theory, we may attribute distinct forms of organization to a subject-matter area. This may be reflected in the cognitive interface of a subject, its presentation. For example, making algebraic concepts applicable to geometric questions affects the organization of geometry; algebraic presentation makes this novel organization ("algebraic structure") cognitively accessible and so relevant. Algebraic structure promotes clarity in geometry, notably by allowing cognitively effective disregard of geometric information irrelevant to questions considered.
