Visualization, Explanation and Reasoning Styles in Mathematics

Abstract

Contributing Authors. P. Mancosu, K.P. Jorgensen and S.A. Pedersen: Introduction. Part I. Mathematical Reasoning And Visualization. P. Mancosu: Visualization in Logic and Mathematics. 1. Diagrams and Images in the Late Nineteenth Century. 2. The Return of the Visual as a Change in Mathematical Style. 3. New Directions of Research and Foundations of Mathematics. Acknowledgements. Notes. References. M. Giaquinto: From Symmetry Perception to Basic Geometry. Introduction. 1. Perceiving a Figure as a Square. 2. A Geometrical Concept for Squares. 3. Getting the Belief. 4. Is It Knowledge? 5. Summary. Notes. References. J.R. Brown: Naturalism, Pictures, and Platonic Intuitions. 1. Naturalism. 2. Platonism. 3. Godel's Platonism. 4. The Concept of Observable. 5. Proofs and Intuitions. 6. Maddy's Naturalism. 7. Refuting the Continuum Hypothesis. Acknowledgements. Appendix: Freiling's 'Philosophical' Refutation of CH. References. M. Giaquinto: Mathematical Activity. 1. Discovery. 2. Explanation. 3. Justification. 4. Refining and Extending the List of Activities. 5. Conc1uding Remarks. Notes. References. Part II. Mathematical Explanation and Proof Styles. J. Hoyrup: Tertium Non Datur: On Reasoning Styles in Early Mathematics. 1. Two Convenient Scapegoats. 2. Old Babylonian Geometric Proto-algebra. 3. Euc1idean Geometry. 4. Stations on the Road. 5. Other Greeks. 6. Proportionality - Reasoning and its Elimination. Notes. References. K. Chemla: The Interplay Between Proof and AIgorithm in 3rd Century China: The Operation as Prescription of Computation and the Operation as Argument. 1. Elements of Context. 2. Sketch of the Proof. 3. First Remarks on the Proof. 4. The Operation as Relation of Transformation. 5. The Essential Link Between Proof and AIgorithm. 6. Conc1usion. Appendix. Notes. References. J. Tappenden: Proof Style and Understanding in Mathematics I:Visualization, Unification and Axiom Choice. 1. Introduction - a 'New Riddle' of Deduction. 2. Understanding and Explanation in Mathematical Methodology: The Target. 3. Understanding, Unification and Explanation - Friedman. 4. Kitcher: Pattems of Argument. 5. Artin and Axiom Choice: 'Visual Reasoning' Without Vision. 6. Summary - the 'new Riddle of Deduction'. Notes. References. J. Hafner and P. Mancosu: The Varieties of Mathematical Explanations. 1. Back to the Facts Themselves. 2. Mathematical Explanation or Explanation in Mathematics? 3. The Search for Explanation within Mathematics. 4. Some Methodological Comments on the General Project. 5. Mark Steiner on Mathematical Explanation. 6. Kummer's Convergence Test. 7. A Test Case for Steiner's Theory. Appendix. Notes. References. R. Netz: The Aesthetics of Mathematics: A Study. 1. The Problem Motivated. 2. Sources of Beauty in Mathematics. 3. Conclusion. Notes. References. Index.