# Explanation and Proof in Mathematics

Hanna, G., Jahnke, H.N., Pulte, H.(eds), Springer, 2010

## 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.

## Table of contents

### 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 NÃºÃ±ez: 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 to Symbols, Evelyne Barbin

Chapter 16. Proof in the Wording: Two modalities from Ancient Chinese Algorithms, Karine Chemla