Explanation and Proof in Mathematics: Philosophical and Educational Perspectives

In the four decades since Imre Lakatos declared mathematics a "quasi-empirical science," increasing attention has been paid to the process of proof and argumentation in the field -- a development paralleled by the rise of computer technology and the mounting interest in the logical underpinnings of mathematics. Explanantion and Proof in Mathematics assembles perspectives from mathematics education and from the philosophy and history of mathematics to strengthen mutual awareness and share recent findings and advances in their interrelated fields. With examples ranging from the geometrists of the 17th century and ancient Chinese algorithms to cognitive psychology and current educational practice, contributors explore the role of refutation in generating proofs, the varied links between experiment and deduction, the use of diagrammatic thinking in addition to pure logic, and the uses of proof in mathematics education (including a critique of "authoritative" versus "authoritarian" teaching styles). A sampling of the coverage: The conjoint origins of proof and theoretical physics in ancient Greece. Proof as bearers of mathematical knowledge. Bridging knowing and proving in mathematical reasoning. The role of mathematics in long-term cognitive development of reasoning. Proof as experiment in the work of Wittgenstein. Relationships between mathematical proof, problem-solving, and explanation. Explanation and Proof in Mathematics is certain to attract a wide range of readers, including mathematicians, mathematics education professionals, researchers, students, and philosophers and historians of mathematics. Part I. Reflections on the Nature and Teaching of Proof Chapter 1. The Conjoint Origin of Proof and Theoretical Physics Hans Niels Jahnke Chapter 2. Lakatos, Lakoff and Nez: Towards a Satisfactory Definition of Continuity. Teun Koetsier Chapter 3. Pre-Axiomatic Mathematical Reasoning: An Algebraic Approach Mary Catherine Leng Chapter 4. Completions, Constructions and Corrollaries Thomas Mormann Chapter 5. Authoritarian vs. Authoritative Teaching: Polya and Lakatos Brendan Larvor Chapter 6. Proofs as Bearers of Mathematical Knowledge Gila Hanna & Ed Barbeau Chapter 7. Mathematicians' Individual Criteria for Accepting Theorems and Proofs: An Empirical Approach Aiso Heinze Part II. Proof and Cognitive Development Chapter 8. Bridging Knowing and Proving in Mathematics: A Didactical Perspective Nicolas Balacheff Chapter 9. The Long-term Cognitive Development of Reasoning and Proof David Tall & Juan Pablo Mejia-Ramos Chapter 10. Historical Artefacts, Semiotic Mediation and Teaching Proof Mariolina Bartolini-Bussi Chapter 11. Proofs, Semiotics and Artefacts of Information Technologies Alessandra Mariotti Part III. Experiments, Diagrams and Proofs Chapter 12. Proof as Experiment in Wittgenstein Alfred Nordmann Chapter 13. Experimentation and Proof in Mathematics Michael D. de Villiers Chapter 14. Proof, Mathematical Problem-Solving, and Explanation in Mathematics Teaching Kazuhiko Nunokawa Chapter 15. Evolving Geometric Proofs in the 17th Century: From Icons toSymbols Evelyne Barbin Chapter 16. Proof in the Wording: Two modalities from Ancient Chinese Algorithms Karine Chemla