Axiomatic Method and Category Theory

Description

This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has.  Read more... Abstract: This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has Content: Preface Acknowledgement Contents Chapter 1: Introduction Part I: A Brief History of the Axiomatic Method Chapter 2: Euclid: Doing and Showing 2.1 Demonstration and ``Monstration'' 2.2 Are Euclid's Proofs Logical? 2.3 Instantiation, Objecthood and Objectivity 2.4 Proto-Logical Deduction and Geometrical Production 2.5 Euclid and Modern Mathematics Chapter 3: Hilbert: Making It Formal 3.1 Foundations of 1899: Logical Form and Mathematical Intuition 3.2 Foundations of 1899: Logicality and Logicism 3.3 Axiomatization of Logic: Logical Form Versus Symbolic Form. 3.4 Foundations of 1927: Intuition Strikes Back3.5 Symbolic Logic and Diagrammatic Logic 3.6 Foundations of 1934-1939: Doing Is Showing? Chapter 4: Formal Axiomatic Method and the Twentieth Century Mathematics 4.1 Set Theory 4.2 Bourbaki 4.3 Galilean Science and Set-Theoretic Foundations of Mathematics 4.4 Towards the New Axiomatic Method: Interpreting Logic Chapter 5: Lawvere: Pursuit of Objectivity 5.1 Elementary Theory of the Category of Sets 5.2 Category of Categories as a Foundation 5.3 Conceptual Theories and Their Presentations. 5.4 Curry-Howard Correspondence and Cartesian Closed Categories5.5 Hyperdoctrines 5.6 Functorial Semantics 5.7 Formal and Conceptual 5.8 Categorical Logic and Hegelian Dialectics 5.9 Toposes and Their Internal Logic Conclusion of Part I Part II: Identity and Categorification Chapter 6: Identity in Classical and Constructive Mathematics 6.1 Paradoxes of Identity and Mathematical Doubles 6.2 Types and Tokens 6.3 Frege and Russell on the Identity of Natural Numbers 6.4 Plato 6.5 Definitions by Abstraction 6.6 Relative Identity 6.7 Internal Relations 6.8 Classes 6.9 Individuals. 6.10 Extension and Intension6.11 Identity in the Intuitionistic Type Theory Chapter 7: Identity Through Change, Category Theory and Homotopy Theory 7.1 Relations Versus Transformations 7.2 How to Think Circle 7.3 Categorification 7.4 Are Identity Morphisms Logical? 7.5 Fibred Categories 7.6 Higher Categories 7.7 Homotopies 7.7.1 Fixing the Associativity 7.7.2 Exploring the Non-associativity 7.8 Model Categories 7.9 Homotopy Type Theory 7.10 Univalent Foundations Conclusion of Part II Part III: Subjective Intuitions and Objective Structures. Chapter 8: How Mathematical Concepts Get Their Bodies8.1 Changing Intuition 8.2 Form and Motion 8.3 Non-Euclidean Intuition 8.4 Lost Ideals 8.5 Are Intuitions Fundamental? Chapter 9: Categories Versus Structures 9.1 Structuralism, Mathematical 9.2 What Replaces What? 9.3 Erlangen Program and Axiomatic Method 9.4 Objective Structures 9.5 Types and Categories of Structures 9.6 Invariance Versus Functoriality 9.7 Are Categories Structures? 9.8 Objects Are Maps Chapter 10: New Axiomatic Method (Instead of Conclusion) 10.1 Unification 10.2 Concentration.