Written by experts in the field, this volume presents a comprehensive investigation into the relationship between argumentation theory and the philosophy of mathematical practice. Argumentation theory studies reasoning and argument, and especially those aspects not addressed, or not addressed well, by formal deduction. The philosophy of mathematical practice diverges from mainstream philosophy of mathematics in the emphasis it places on what the majority of working mathematicians actually do, rather than on mathematical foundations.
The book begins by first challenging the assumption that there is no role for informal logic in mathematics. Next, it details the usefulness of argumentation theory in the understanding of mathematical practice, offering an impressively diverse set of examples, covering the history of mathematics, mathematics education and, perhaps surprisingly, formal proof verification. From there, the book demonstrates that mathematics also offers a valuable testbed for argumentation theory. Coverage concludes by defending attention to mathematical argumentation as the basis for new perspectives on the philosophy of mathematics.
Table of Contents
Introduction.- Part I. What are Mathematical Arguments?.- Chapter 1. Non-Deductive Logic in Mathematics: The Probability of Conjectures; James Franklin.- Chapter 2. Arguments, Proofs, and Dialogues; Erik C. W. Krabbe.- Chapter 3. Argumentation in Mathematics; Jesu's Alcolea Banegas.- Chapter 4. Arguing Around Mathematical Proofs; Michel Dufour.- Part II. Argumentation as a Methodology for Studying Mathematical Practice.- Chapter 5. An Argumentative Approach to Ideal Elements in Mathematics; Paola Cantù.- Chapter 6. How Persuaded Are You? A Typology of Responses; Matthew Inglis and Juan Pablo Mejía-Ramos.- Chapter 7. Revealing Structures of Argumentations in Classroom Proving Processes; Christine Knipping and David Reid.- Chapter 8. Checking Proofs; Jesse Alama and Reinhard Kahle.- Part III. Mathematics as a Testbed for Argumentation Theory.- Chapter 9. Dividing by Zeroand Other Mathematical Fallacies; Lawrence H. Powers.- Chapter 10. Strategic Maneuvering in Mathematical Proofs; Erik C. W. Krabbe.- Chapter. 11 Analogical Arguments in Mathematics; Paul Bartha.- Chapter 12. What Philosophy of Mathematical Practice Can Teach Argumentation Theory about Diagrams and Pictures; Brendan Larvor.- Part IV. An Argumentational Turn in the Philosophy of Mathematics.- Chapter 13. Mathematics as the Art of Abstraction; Richard L. Epstein.- Chapter 14. Towards a Theory of Mathematical Argument; Ian J. Dove.- Chapter 15. Bridging the Gap Between Argumentation Theory and the Philosophy of Mathematics; Alison Pease, Alan Smaill, Simon Colton and John Lee.- Chapter 16. Mathematical Arguments and Distributed Knowledge; Patrick Allo, Jean Paul Van Bendegem and Bart Van Kerkhove.- Chapter 17. The Parallel Structure of Mathematical Reasoning; Andrew Aberdein.- Index.