What is a mathematical concept?

Description

Cover; Half-title; Title page; Copyright information; Table of contents; List of images; Notes on contributors; Introduction; The Context for This Book: Philosophy and Cognition; Themes and Chapters; On Reading the Book; References; Part I; 1 Of Polyhedra and Pyjamas: Platonism and Induction in Meaning-Finitist Mathematics; Introduction; Strong Program Meaning Finitism; Induction; Platonism; Conclusion; Acknowledgments; References; 2 Mathematical Concepts? The View from Ancient History; Introduction; Infinity, Actually; Indeed, It's Not That They Couldn't; What We Share; References; Part II.;Leading thinkers in mathematics, philosophy and education offer new insights into the fundamental question: what is a mathematical concept? Cover Half-title Title page Copyright information Table of contents List of images Notes on contributors Introduction The Context for This Book: Philosophy and Cognition Themes and Chapters On Reading the Book References Part I 1 Of Polyhedra and Pyjamas: Platonism and Induction in Meaning-Finitist Mathematics Introduction Strong Program Meaning Finitism Induction Platonism Conclusion Acknowledgments References 2 Mathematical Concepts? The View from Ancient History Introduction Infinity, Actually Indeed, It's Not That They Couldn't What We Share References Part II. 3 Notes on the Syntax and Semantics Distinction, or Three Moments in the Life of the Mathematical DrawingIntroduction Two Subgenres First Moment: The Euclidean Diagram Second Moment: What Is the Hand? Third Moment: The Incidental Drawing Conclusion Acknowledgements References 4 Concepts as Generative Devices Proposition 1: Concepts Are Not Merely Metaphors or Representations Proposition 2: Concepts Are Not Mental Constructs Abstracted from the Material World Proposition 3: Concepts Are Vibrant and Indeterminate, Having One Foot in the Virtual and One in the Actual. Proposition 4: Concepts Operate as Both Logical and Ontological DevicesProposition 5: There Is No a priori Logical Ordering between Mathematical Concepts Proposition 6: Concepts Emerge from Aesthetico-Political Acts Concluding Comments References Part III 5 Bernhard Riemann's Conceptual Mathematics and the Pedagogy of Mathematical Concepts Introduction: From Conceptual Philosophy to Conceptual Mathematics Planes of Immanence and the Architecture of Concepts The Concept of Manifold: Space and Geometry in Riemann Conclusion: A Pedagogy of the Concept References. 6 Deleuze and the Conceptualisable Character of Mathematical TheoriesThe Theory of Mathematical Problems and the History of Mathematics Lautman, Cavaillès and the Mathematical Real Galois (Formal) or Poincaré (Informal) on the Mathematical Expression of Problems? References Part IV 7 Homotopy Type Theory and the Vertical Unity of Concepts in Mathematics Introduction Varieties of Concept Addition, Multiplication and Exponentiation Groupoids, Identity and Symmetry Spaces and Duality Conclusion References 8 The Perfectoid Concept: Test Case for an Absent Theory Perfectoid Prologue. Starting PointsScholze's Perfectoid Concept The Concept's Reception Discussion References Part V 9 Queering Mathematical Concepts What Are Mathematical Concepts? Looking Awry at Mathematical Concepts, or What Do Mathematical Concepts Do? Queering Mathematical Concepts, or What Could Mathematical Concepts Do? Where Do We Go from Here? Acknowledgements References 10 Mathematics Concepts in the News Reporting Mathematics Education Theoretical Background Framing Constructing Concepts Conclusions References 11 Concepts and Commodities in Mathematical Learning.