The Role of Concepts in Mathematical Knowledge
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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.
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{'Code': '1353', 'Text': 'Hist & Philosophy of SET'}
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