Logos and Alogon: Thinkable and the Unthinkable in Mathematics, from the Pythagoreans to the Moderns

Description

This book is a philosophical study of mathematics, pursued by considering and relating two aspects of mathematical thinking and practice, especially in modern mathematics, which, having emerged around 1800, consolidated around 1900 and extends to our own time, while also tracing both aspects to earlier periods, beginning with the ancient Greek mathematics. The first aspect is conceptual, which characterizes mathematics as the invention of and working with concepts, rather than only by its logical nature. The second, Pythagorean, aspect is grounded, first, in the interplay of geometry and algebra in modern mathematics, and secondly, in the epistemologically most radical form of modern mathematics, designated in this study as radical Pythagorean mathematics. This form of mathematics is defined by the role of that which beyond the limits of thought in mathematical thinking, or in ancient Greek terms, used in the book’s title, an alogon in the logos of mathematics. The outcome of this investigation is a new philosophical and historical understanding of the nature of modern mathematics and mathematics in general. The book is addressed to mathematicians, mathematical physicists, and philosophers and historians of mathematics, and graduate students in these fields. Preface Acknowledgments Contents Chapter 1: Introduction 1.1 Prologue: Logos and Alogon in the Tragic Age of the Greeks 1.2 Modern Mathematics and Radical Pythagorean Mathematics: Abstraction and Alogon 1.3 Thinking Mathematical Thinking: From History to Philosophy References Chapter 2: The Spirit of Pythagoreans Against Platonism: From Logos to Alogon, and from Alogon to Logos, in Mathematical Think... 2.1 Introduction 2.2 Radical Pythagorean Mathematics and the Nature of Reality 2.3 What Is a Mathematical Concept? 2.4 Modern Mathematics Between Geometry and Algebra 2.5 Radical Pythagorean Thinking 2.6 Conclusion References Chapter 3: ``Comprehending the Connection of Things´´: Bernhard Riemann and Conceptual Thinking in Mathematics 3.1 Introduction 3.2 Philosophy of Spatiality, from Kant to Riemann and Beyond 3.3 Manifolds, Spaces, and Geometries 3.4 Space-Time-Matter 3.5 Conclusion References Chapter 4: What Is a Curve?: A Pythagorean Essay 4.1 Introduction 4.2 Curves as Algebra: Descartes/Fermat/Diophantus 4.3 Curves as Surfaces, Surfaces as Curves: Riemann/Riemann/Riemann 4.4 Curves as Discrete Manifolds: Grothendieck/Weil/Riemann 4.5 Conclusion References Chapter 5: ``To Create More Worlds´´: Mathematical Practice as Philosophy 5.1 Introduction 5.2 Algebra: Abel and Galois 5.3 Geometry: Lobachevsky and Riemann 5.4 Algebraic Geometry: Weil and Grothendieck 5.5 Conclusion: Technology Against Ontology in Mathematical Practice as Philosophy References Chapter 6: Who Thinks Abstractly?: From Modern Geometry to Modern Algebra with Emmy Noether 6.1 Introduction 6.2 Philosophy (with Mathematics): Abstract Thinking, Algebraic and Geometrical, in Mathematics 6.3 Geometry (with Algebra): Noether´s Theorems and Noether´s Noether´s Theorems 6.4 Algebra (with Algebra): Abstract Algebra from Galois to Noether 6.5 Topology (with Algebra): From Structures to Morphisms 6.6 Conclusion References Chapter 7: Physics as Modern Mathematics: From Relativity to Quantum Theory, and Beyond 7.1 Introduction 7.2 Modern Mathematics in Relativity and Quantum Theory 7.3 ``The Heisenberg Method´´: Algebra and Alogon 7.4 Matrices, Spinors, and Group Representations: Algebra, Geometry, and Symmetry in Quantum Theory 7.5 Platonic Solids, Galois Atoms, and the Cosmic Galois Group 7.6 From Physics to Mathematics: The Yang-Mills Theory and Differential Topology 7.7 Conclusion References Name Index Subject Index