How We Understand Mathematics. Conceptual Integration in the Language of Mathematical Description

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Acknowledgments Contents Chapter 1: Introduction 1.1 The Effectiveness of Mathematics, Conceptual Integration, and Small Spatial Stories 1.2 The Point and Method of the Book 1.3 Who Is the Book Addressed To 1.4 The Organization of the Book Chapter 2: The Theoretical Framework and the Subject of Study 2.1 Overview 2.2 Language, Cognition, and Conceptual Integration 2.2.1 Cognitive Linguistics 2.2.2 Conceptual Integration (Blending) Theory: The Basic Architecture 2.2.3 The Criticism of the Conceptual Integration Theory 2.2.4 The Constitutive and Governing Principles 2.2.5 Small Spatial Stories and Image Schemas 2.3 Modern Algebra for Beginners Chapter 3: Sets 3.1 Overview 3.2 The Primitive Notions: Set and an Element 3.3 Subsets and Equal Sets 3.3.1 Subsets 3.3.2 Equal Sets 3.4 The Null Set 3.5 The Union of Sets 3.6 The Intersection of Sets 3.7 Image Schemas and Small Spatial Stories for Sets and Elements 3.8 Defining Sets with a Condition and Russell’s Paradox 3.9 Proposition, Proof, and Small Spatial Stories Chapter 4: Mappings 4.1 Overview 4.2 The Mapping as “a Carrier” 4.3 The “Rigorous” Definition, Ordered Pairs 4.4 Circularity of the “Rigorous” Definition and Conceptual Integration 4.5 He Small Spatial Story of the Matchmaker 4.6 Definition by Graph, the Small Spatial Story of a Hiker 4.7 Structured Small Spatial Stories vs. Circularity of the Definition Chapter 5: Groups 5.1 Overview 5.2 The Definition of a Group and the Story of the Matchmakers 5.3 Abelian Groups, Finite Groups, and the Beauty of Mathematics (Part 1) 5.4 On the Objective Nature of Mathematics 5.5 The Uniqueness of the Group Elements and Conceptual Blending 5.6 The Force Dynamics of Mathematical Proof 5.7 The Subgroups, Lagrange’s Theorem, and the Beauty of Mathematics (Part 2) 5.8 Normal Subgroups and the Beauty of Mathematics (Part 3) 5.9 The Homomorphism 5.9.1 Homomorphism and the Carrier Story 5.9.2 Homomorphism and the Matchmaker Story 5.9.3 Homomorphism and the Beauty of Mathematics (Part 4) Chapter 6: Rings, Fields, and Vector Spaces 6.1 Overview 6.2 Definition of a Ring, Small Spatial Story of Three Matchmakers 6.3 The Structure of the Ring 6.3.1 How the Ring Matchmakers Cooperate 6.3.2 Ring as a Closed Container, Force Dynamics of Proof 6.4 Rings, Fields, and Arithmetic 6.5 From Set and Element to Arithmetic: The Story So Far 6.6 Multiplication by Zero Equals Zero: Proof as an Actor 6.7 Small Spatial Stories of Addition and Subtraction 6.7.1 The Small Spatial Story of Jenga Blocks 6.7.2 Cayley’s Theorem 6.7.3 The Small Spatial Story of Three Bricks 6.7.4 From Bricks to Arithmetic 6.7.5 Arithmetic at School vs. Modern Algebra 6.8 Vector Space and the Seven Matchmakers Chapter 7: Summary and Conclusion Bibliography