Deleuze and the history of mathematics : in defence of the 'new'

Gilles Deleuze's engagements with mathematics, replete in his work, rely upon the construction of alternative lineages in the history of mathematics, which challenge some of the self imposed limits that regulate the canonical concepts of the discipline. For Deleuze, these challenges provide an opportunity to reconfigure particular philosophical problems - for example, the problem of individuation - and to develop new concepts in response to them. The highly original research presented in this book explores the mathematical construction of Deleuze's philosophy, as well as addressing the undervalued and often neglected question of the mathematical thinkers who influenced his work. In the wake of Alain Badiou's recent and seemingly devastating attack on the way the relation between mathematics and philosophy is configured in Deleuze's work, Simon B.Duffy offers a robust defence of the structure of Deleuze's philosophy and, in particular, the adequacy of the mathematical problems used in its construction. By reconciling Badiou and Deleuze's seemingly incompatible engagements with mathematics, Duffy succeeds in presenting a solid foundation for Deleuze's philosophy, rebuffing the recent challenges against it. Cover-Page 1 Half-Title 2 Series 3 Title 5 Copyright 6 Dedication 8 Contents 9 Acknowledgments 10 List of Abbreviations 12 Introduction 13 1 Leibniz and the Concept of the Infinitesimal 18 Leibniz’s law of continuity and the infinitesimal calculus 19 Newton’s method of fluxions and infinite series 22 The emergence of the concept of the function 24 Subsequent developments in mathematics: The problem of rigor 25 The theory of singularities 26 The characteristics of a point-fold as reflected in the point of inflection 28 Subsequent developments in mathematics: Weierstrass and Poincaré 31 The development of a differential philosophy 34 The qualitative theory of differential equations 38 Deleuze’s “Leibnizian” interpretation of the theory of compossibility 42 Point of view and the theory of the differential unconscious 44 The mathematical representation of matter, motion, and the continuum 45 The Koch curve and the folded tunic: The fractal nature of motion 48 The metaphysics of monads, and bodies as “well-founded phenomena” 50 Deleuze’s characterization of Leibniz’s account of matter 51 Overcoming the limits of Leibniz’s metaphysics 53 Spinoza and the logic of different/ciation 55 2 Maimon’s Critique of Kant’s Approach to Mathematics 58 Kant on the construction of mathematical concepts in pure intuition 58 The concept of the straight line 63 Maimon’s critique of Kant 66 Maimonic reduction 71 The laws of sensibility 73 Noumena, phenomena, and regulative ideas 83 Bordas-Demoulin on the differential relation as “the universal function” 88 Maimon’s infinite intellect is displaced by a theory of problems 90 The rigorous algorithm of Wronski’s transcendental philosophy 91 A “problematic” is “the ensemble of the problem and its conditions” 92 Abel and Galois on the question of the solvability of polynomial equations 95 3 Bergson and Riemann on Qualitative Multiplicity 101 The role of judgment in the determination of the idea of an extensive magnitude 101 Mechanical explanation as a method or as a doctrine? 105 Bergson’s problem with the cinematographical method overcome 112 The Riemannian concept of multiplicity and the Dedekind cut 113 Deleuze’s rehabilitation and extension of Bergson’s project 119 4 Lautman’s Concept of the Mathematical Real 128 Lautman’s axiomatic structuralism 128 The metaphysics of logic: A philosophy of mathematical genesis 131 Lautman’s Platonism 134 Problematic ideas and the concept of genesis 136 Heidegger and the naive period in the history of mathematical logic 138 The virtual in Lautman 140 Deleuze and the calculus of problems 141 The logic of the calculus of problems 144 5 Badiou and Contemporary Mathematics 147 Badiou and the role of mathematics as ontology 149 Orthodox Platonism in mathematics and its problems 151 Badiou’s “modern Platonist” response and its reformulation of the question 154 Cantor’s account of transfinite numbers or ordinals 156 The Platonist implications of axiomatic set theory 158 The model-theoretic implications of Badiou’s “modern Platonism” 161 On the difference between set theory and category theory 164 Mathematics as ontology in Badiou and Deleuze 165 Conclusion 171 The “vindication” of Leibniz’s account of the differential 171 The scienticity debate in Deleuze studies 178 Badiou’s relation to Lautman and the mathematical real 179 Notes 185 Introduction 185 Chapter 1 185 Chapter 2 190 Chapter 3 194 Chapter 4 196 Chapter 5 198 Conclusion 201 Bibliography 203 Index 215 "Gilles Deleuze's engagements with mathematics, replete in his work, rely upon the construction of alternative lineages in the history of mathematics, which challenge some of the self imposed limits that regulate the canonical concepts of the discipline. For Deleuze, these challenges are an opportunity to reconfigure particular philosophical problems - for example, the problem of individuation - and to develop new concepts in response to them. The highly original research presented in this book explores the mathematical construction of Deleuze's philosophy, as well as addressing the undervalued and often neglected question of the mathematical thinkers who influenced his work. In the wake of Alain Badiou's recent and seemingly devastating attack on the way the relation between mathematics and philosophy is configured in Deleuze's work, Simon Duffy offers a robust defence of the structure of Deleuze's philosophy and, in particular, the adequacy of the mathematical problems used in its construction. By reconciling Badiou and Deleuze's seeming incompatible engagements with mathematics, Duffy succeeds in presenting a solid foundation for Deleuze's philosophy, rebuffing the recent challenges against it."--Bloomsbury Publishing. Acknowledgements \ List of Abbreviations \ Introduction \ 1. Leibniz and the Concept of the Infinitesimal \ 2. Maimon's Critique of Kant's Approach to Mathematics \ 3. Bergson and Riemann on Qualitative Multiplicity \ 4. Lautman's Concept of the Mathematical Real \ 5. Badiou and Contemporary Mathematics \ Conclusion \ Notes \ Bibliography \ Index Machine Generated Contents Note: 1. Leibniz And The Concept Of The Infinitesimal -- 2. Maimon's Critique Of Kant's Approach To Mathematics -- 3. Bergson And Riemann On Qualitative Multiplicity -- 4. Lautman's Concept Of The Mathematical Real -- 5. Badiou And Contemporary Mathematics. Simon B. Duffy. Includes Bibliographical References And Index.