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.
Author(s): Gila Hanna, Hans Niels Jahnke, Helmut Pulte (auth.), Gila Hanna, Hans Niels Jahnke, Helmut Pulte (eds.)
Edition: 1
Publisher: Springer US
Year: 2010
Language: English
Pages: 294
Tags: Mathematics Education; Philosophy of Education
Front Matter....Pages i-viii
Front Matter....Pages 16-16
Introduction....Pages 1-13
The Conjoint Origin of Proof and Theoretical Physics....Pages 17-32
Lakatos, Lakoff and Núñez: Towards a Satisfactory Definition of Continuity....Pages 33-46
Preaxiomatic Mathematical Reasoning: An Algebraic Approach....Pages 47-57
Completions, Constructions, and Corollaries....Pages 59-70
Authoritarian Versus Authoritative Teaching: Polya and Lakatos....Pages 71-83
Proofs as Bearers of Mathematical Knowledge....Pages 85-100
Mathematicians’ Individual Criteria for Accepting Theorems and Proofs: An Empirical Approach....Pages 101-111
Front Matter....Pages 114-114
Bridging Knowing and Proving in Mathematics: A Didactical Perspective....Pages 115-135
The Long-Term Cognitive Development of Reasoning and Proof....Pages 137-149
Historical Artefacts, Semiotic Mediation and Teaching Proof....Pages 151-167
Proofs, Semiotics and Artefacts of Information Technologies....Pages 169-188
Front Matter....Pages 190-190
Proof as Experiment in Wittgenstein....Pages 191-204
Experimentation and Proof in Mathematics....Pages 205-221
Proof, Mathematical Problem-Solving, and Explanation in Mathematics Teaching....Pages 223-236
Evolving Geometric Proofs in the Seventeenth Century: From Icons to Symbols....Pages 237-251
Proof in the Wording: Two Modalities from Ancient Chinese Algorithms....Pages 253-285
Back Matter....Pages 287-294
