The Argument of Mathematics (Logic, Epistemology, and the Unity of Science Book 30)

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. ​ 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. Machine Generated Contents Note: 1. Introduction / Ian J. Dove -- Pt. I What Are Mathematical Arguments? -- 2. Non-deductive Logic In Mathematics: The Probability Of Conjectures / James Franklin -- 3. Arguments, Proofs, And Dialogues / Erik C.w. Krabbe -- 4. Argumentation In Mathematics / Jesus Alcolea Banegas -- 5. Arguing Around Mathematical Proofs / Michel Dufour -- Pt. Ii Argumentation As A Methodology For Studying Mathematical Practice -- 6. An Argumentative Approach To Ideal Elements In Mathematics / Paola Cantu -- 7. How Persuaded Are You? A Typology Of Responses / Juan Pablo Mejia-ramos -- 8. Revealing Structures Of Argumentations In Classroom Proving Processes / David Reid -- 9. Checking Proofs / Reinhard Kahle -- Pt. Iii Mathematics As A Testbed For Argumentation Theory -- 10. Dividing By Zero -- And Other Mathematical Fallacies / Lawrence H. Powers -- 11. Strategic Maneuvering In Mathematical Proofs / Erik C.w. Krabbe -- 12. Analogical Arguments In Mathematics / Paul Bartha -- 13. What Philosophy Of Mathematical Practice Can Teach Argumentation Theory About Diagrams And Pictures / Brendan Larvor -- Pt. Iv An Argumentational Turn In The Philosophy Of Mathematics -- 14. Mathematics As The Art Of Abstraction / Richard L. Epstein -- 15. Towards A Theory Of Mathematical Argument / Ian J. Dove -- 16. Bridging The Gap Between Argumentation Theory And The Philosophy Of Mathematics / John Lee -- 17. Mathematical Arguments And Distributed Knowledge / Bart Van Kerkhove -- 18. The Parallel Structure Of Mathematical Reasoning / Andrew Aberdein. Andrew Aberdein, Ian J. Dove, Editors. Includes Bibliographical References And Index. 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; Jesús 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 Zero—and 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. Front Matter....Pages i-x Introduction....Pages 1-8 Front Matter....Pages 9-9 Non-deductive Logic in Mathematics: The Probability of Conjectures....Pages 11-29 Arguments, Proofs, and Dialogues....Pages 31-45 Argumentation in Mathematics....Pages 47-60 Arguing Around Mathematical Proofs....Pages 61-76 Front Matter....Pages 77-77 An Argumentative Approach to Ideal Elements in Mathematics....Pages 79-99 How Persuaded Are You? A Typology of Responses....Pages 101-117 Revealing Structures of Argumentations in Classroom Proving Processes....Pages 119-146 Checking Proofs....Pages 147-170 Front Matter....Pages 171-171 Dividing by Zero—and Other Mathematical Fallacies....Pages 173-179 Strategic Maneuvering in Mathematical Proofs....Pages 181-197 Analogical Arguments in Mathematics....Pages 199-237 What Philosophy of Mathematical Practice Can Teach Argumentation Theory About Diagrams and Pictures....Pages 239-253 Front Matter....Pages 255-255 Mathematics as the Art of Abstraction....Pages 257-289 Towards a Theory of Mathematical Argument....Pages 291-308 Bridging the Gap Between Argumentation Theory and the Philosophy of Mathematics....Pages 309-338 Mathematical Arguments and Distributed Knowledge....Pages 339-360 The Parallel Structure of Mathematical Reasoning....Pages 361-380 Back Matter....Pages 381-393