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Index
Cover
Halftitle
Also available
Title page
Copyright
Contents
Translator’s Note
Acknowledgements
Introduction, by Jacques Lautman
Secondary Bibliography on the Work of Albert Lautman
Albert Lautman and the Creative Dialectic of Modern Mathematics, by Fernando Zalamea
1. Effective Mathematics
2. Structure and Unity
3. Mixes
4. Notions and Ideas
5. Platonism
6. Category Theory
Preface to the 1977 Edition, by Jean Dieudonné
Considerations on Mathematical Logic
Mathematics and Reality
International Congress of the Philosophy of Science
On the Reality Inherent to Mathematical Theories
The Axiomatic and the Method of Division
1. Equality
2. Multiplication
3. Unity
4. Measure and the Integral
5. The Absolute Value
Book I: Essay on the Unity of the Mathematical Sciences in their Current Development
Introduction: Two Kinds of Mathematics
Chapter 1 The Structure of a Domain of Magnitudes and the Decomposition of Its Elements: Dimensional Considerations in Analysis
Chapter 2 The Domain and Numbers: Non-Euclidean Metrics in the Theory of Analytic Functions
Chapter 3 The Algebra of Non-Commutative Magnitudes: Pfaffian Forms and the Theory of Differential Equations
Chapter 4 The Continuous and the Discontinuous: Analysis and the Theory of Numbers
Conclusion
Book II: Essay on the Notions of Structure and Existence in Mathematics
Introduction: On the Nature of the Real in Mathematics
Section 1: The Schemas of Structure
Chapter 1 The Local and the Global
1. Differential Geometry and Topology
2. The Theory of Closed Groups
3. Approximate Representation of Functions
Chapter 2 Intrinsic Properties and Induced Properties
1. Parallelism on a Riemann Manifold
2. Structural Properties and Situational Properties in Algebraic Topology
3. Duality Theorems
4. The Limitations of Reduction
Chapter 3 The Ascent towards the Absolute
1. Galois’s Theory
2. Class Field Theory
3. The Universal Covering Surface
4. The Uniformization of Algebraic Functions on a Riemann Surface
Section 2: The Schemas of Genesis
Chapter 4 Essence and Existence
1. The Problems of Mathematical Logic
2. Existence Theorems in the Theory of Algebraic Functions
3. Existence Theorems in Class Field Theory
4. The Theory of the Representation of Groups
Chapter 5 ‘Mixes’
1. Hilbert Space
2. Normal Families of Analytic Functions
Chapter 6 On the Exceptional Character of Existence
1. The Methods of Poincaré
2. The Singularities of Analytic Functions
Conclusion
Book III New Research on the Dialectical Structure of Mathematics
Foreword
Chapter 1 The Genesis of the Entity from the Idea
1. The Genesis of Mathematics from the Dialectic
Chapter 2 The Analytic Theory of Numbers
1. The Law of Reciprocity
2. The Distribution of Primes and the Measurement of the Increase to Infinity
3. Conclusion
Letter to Mathematician Maurice Fréchet
Book IV Symmetry and Dissymmetry in Mathematics and Physics
Chapter 1 Physical Space
Chapter 2 The Problem of Time
1. Sensible Time and Mathematical Physics
2. The Theory of Partial Differential Equations
3. The Theory of Differential Equations and Topology
Notes
Bibliography
Index
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