Visualization, explanation and reasoning styles in mathematics: [based on the Meeting "Mathematics as Rational Activity" held at Roskilde University, Denmark, from November 1 to November 3, 2001]

In the 20th century philosophy of mathematics has to a great extent been dominated by views developed during the so-called foundational crisis in the beginning of that century. These views have primarily focused on questions pertaining to the logical structure of mathematics and questions regarding the justi?cation and consistency of mathematics. Paradigmatic in this - spect is Hilbert’s program which inherits from Frege and Russell the project to formalize all areas of ordinary mathematics and then adds the requi- ment of a proof, by epistemically privileged means (?nitistic reasoning), of the consistency of such formalized theories. While interest in modi?ed v- sions of the original foundational programs is still thriving, in the second part of the twentieth century several philosophers and historians of mat- matics have questioned whether such foundational programs could exhaust the realm of important philosophical problems to be raised about the nature of mathematics. Some have done so in open confrontation (and hostility) to the logically based analysis of mathematics which characterized the cl- sical foundational programs, while others (and many of the contributors to this book belong to this tradition) have only called for an extension of the range of questions and problems that should be raised in connection with an understanding of mathematics. The focus has turned thus to a consideration of what mathematicians are actually doing when they produce mathematics. Questions concerning concept-formation, understanding, heuristics, changes instyle of reasoning, the role of analogies and diagrams etc. Content: 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. Introduction / -- P. Mancosu ... [et al.] -- PART I. MATHEMATICAL REASONING AND VISUALIZATION Introduction / -- P. Mancosu ... [et al.] -- PART I. MATHEMATICAL REASONING AND VISUALIZATION -- P. Mancosu Introduction / -- P. Mancosu ... [et al.] -- PART I. MATHEMATICAL REASONING AND VISUALIZATION -- P. Mancosu -- M. Giaquinto Introduction / -- P. Mancosu ... [et al.] -- PART I. MATHEMATICAL REASONING AND VISUALIZATION -- P. Mancosu -- M. Giaquinto -- J.R. Brown Introduction / -- P. Mancosu ... [et al.] -- PART I. MATHEMATICAL REASONING AND VISUALIZATION -- P. Mancosu -- M. Giaquinto -- J.R. Brown -- M. Giaquinto -- PART II. MATHEMATICAL EXPLANATION AND PROOF STYLES Introduction / -- P. Mancosu ... [et al.] -- PART I. MATHEMATICAL REASONING AND VISUALIZATION -- P. Mancosu -- M. Giaquinto -- J.R. Brown -- M. Giaquinto -- PART II. MATHEMATICAL EXPLANATION AND PROOF STYLES -- J. Høyrup Introduction / -- P. Mancosu ... [et al.] -- PART I. MATHEMATICAL REASONING AND VISUALIZATION -- P. Mancosu -- M. Giaquinto -- J.R. Brown -- M. Giaquinto -- PART II. MATHEMATICAL EXPLANATION AND PROOF STYLES -- J. Høyrup -- K. Chemla Introduction / -- P. Mancosu ... [et al.] -- PART I. MATHEMATICAL REASONING AND VISUALIZATION -- P. Mancosu -- M. Giaquinto -- J.R. Brown -- M. Giaquinto -- PART II. MATHEMATICAL EXPLANATION AND PROOF STYLES -- J. Høyrup -- K. Chemla -- J. Tappenden Introduction / -- P. Mancosu ... [et al.] -- PART I. MATHEMATICAL REASONING AND VISUALIZATION -- P. Mancosu -- M. Giaquinto -- J.R. Brown -- M. Giaquinto -- PART II. MATHEMATICAL EXPLANATION AND PROOF STYLES -- J. Høyrup -- K. Chemla -- J. Tappenden -- J. Hafner, P. Mancosu Introduction /-- P. Mancosu ... [et al.]-- PART I. MATHEMATICAL REASONING AND VISUALIZATION-- P. Mancosu-- M. Giaquinto-- J.R. Brown-- M. Giaquinto-- PART II. MATHEMATICAL EXPLANATION AND PROOF STYLES-- J. Høyrup-- K. Chemla-- J. Tappenden-- J. Hafner, P. Mancosu-- R. Netz. Introduction / P. Mancosu [and Others] -- Part I. Mathematical Reasoning And Visualization -- Visualization In Logic And Mathematics / P. Mancosu -- From Symmetry Perception To Basic Geometry / M. Giaquinto -- Naturalism, Pictures, And Platonic Intuitions / J.r. Brown -- Mathematical Activity / M. Giaquinto -- Part Ii. Mathematical Explanation And Proof Styles -- Tertium Non Datur: On Reasoning Styles In Early Mathematics / J. Høyrup -- The Interplay Between Proof And Algorithm In 3rd Century China: The Operation As Prescription Of Computation And The Operation As Argument / K. Chemla -- Proof Style And Understanding In Mathematics I: Visualization, Unification And Axiom Choice / J. Tappenden -- The Varieties Of Mathematical Explanations / J. Hafner, P. Mancosu -- The Aesthetics Of Mathematics: A Study / R. Netz. Edited By Paolo Mancosu, Klaus Frovin Jørgensen And Stig Andur Pedersen. Includes Bibliographical References And Index. "This book contains groundbreaking contributions to the philosophical analysis of mathematical practice. The reader will find here systematic philosophical analyses as well as a wealth of philosophically informed case studies ranging from Babylonian, Greek, and Chinese mathematics to nineteenth century real and complex analysis."--Jacket