Phenomenology, Logic, and the Philosophy of Mathematics

Offering a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics, this 2005 book is divided into three parts. Part I contains a general essay on Husserl's conception of science and logic, an essay of mathematics and transcendental phenomenology, and an essay on phenomenology and modern pure geometry. Part II is focused on Kurt Godel's interest in phenomenology. It explores Godel's ideas and also some work of Quine, Penelope Maddy and Roger Penrose. Part III deals with elementary, constructive areas of mathematics. These are areas of mathematics that are closer to their origins in simple cognitive activities and in everyday experience. This part of the book contains essays on intuitionism, Hermann Weyl, the notion of constructive proof, Poincaré and Frege. Half-title......Page 3 Dedication......Page 4 Title......Page 5 Copyright......Page 6 Contents......Page 7 Acknowledgments......Page 9 Introduction: Themes and Issues......Page 13 §1......Page 14 §2......Page 22 PART I REASON, SCIENCE, AND MATHEMATICS......Page 31 1 Science as a Triumph of the Human Spirit and Science in Crisis: Husserl and the Fortunes of Reason......Page 33 §1 Arithmetic, Geometry, Logic, and the Science of All Possible Sciences......Page 36 §2 Phenomenological Philosophy and the Foundation of the Sciences......Page 46 §3 The Crisis of the Modern Sciences......Page 51 §4 Conclusion......Page 56 §1 A Précis of Problems in the Philosophy of Mathematics......Page 58 §2 The Background of Husserl's View: Objectivity and Subjectivity in Mathematics......Page 62 §3 Intentionality......Page 63 §4 Mathematical Objects......Page 67 §5 Knowledge of Mathematical Objects......Page 70 §6 Against Fictionalism......Page 76 §7 Descriptive Clarification of the Meaning of Mathematical Concepts......Page 77 §8 Conclusion......Page 79 3 Free Variation and the Intuition of Geometric Essences: Some Reflections on Phenomenology and Modern Geometry......Page 81 §1 Ideation in Geometry: Some Examples......Page 82 §2 Hierarchies of Geometric Essences......Page 88 §3 Spatial Ontologies and Unified, A Priori Science......Page 92 §4 The Origins of Geometry......Page 95 §5 Applications of Ideation Outside Geometry......Page 100 PART II KURT GÖDEL, PHENOMENOLOGY, AND THE PHILOSOPHY OF MATHEMATICS......Page 103 4 Kurt Gödel and Phenomenology......Page 105 §1......Page 106 §2......Page 113 §3......Page 122 Postscript (1992)......Page 123 5 Gödel's Philosophical Remarks on Logic and Mathematics......Page 124 6 Gödel’s Path from the Incompleteness Theorems (1931) to Phenomenology (1961)......Page 137 §1 Some Ideas from Phenomenology......Page 138 §2 'Leftward' and 'Rightward' Viewpoints in Philosophy......Page 142 §3 The Incompleteness Theorems and Hilbert's Program......Page 143 §4 The Incompleteness Theorems and Carnap's Program (Gödel *1953/59)......Page 148 §5 Against the Elimination of Rational Intuition......Page 150 §6 From the Incompleteness Theorems to Phenomenology......Page 153 §7 Why Phenomenology?......Page 158 7 Gdel and the Intuition of Concepts......Page 161 §1 From Concrete Signs to Abstract Concepts......Page 162 §2 The Place of Concepts in Husserlian Phenomenology: Intentionality and Objects of Experience......Page 167 §3 Some Examples of How Concepts Function in Our Experience......Page 169 §4 Some Relations of Concepts and the Space of Concepts......Page 174 §5 Meaning Clarification......Page 175 §6 The Phenomenological Ontology of Concepts......Page 179 §7 Concepts Are Not Subjective Ideas......Page 182 §8 Conceptual Intuition and the Alleged Problem of 'Epistemic Contact'......Page 183 §9 Some Positions That Ignore or Seek to Eliminate the Intuition of Abstract Concepts......Page 184 §10 Conclusion......Page 188 8 Gödel and Quine on Meaning and Mathematics......Page 189 §1 The Call for Meaning Clarification (1961)......Page 190 §2 Meaning Clarification and Reductionism......Page 193 §3 The Contrast with Quine......Page 195 §4 Analyticity......Page 197 §5 Rational Intuition and Analyticity......Page 200 §6 Mathematical Content and Theoretical Hypotheses of Natural Science......Page 202 §7 Application of Ideas on Incompleteness, Consistency, and Solvability......Page 204 §8 Mathematical Content and Quine's Conception of Set Theory......Page 206 §9 Mathematical Content and Unapplied Parts of Mathematics......Page 207 §11 Conclusion......Page 211 9 Maddy on Realism in Mathematics......Page 213 10 Penrose on Minds and Machines......Page 227 PART III CONSTRUCTIVISM, FULFILLABLE INTENTIONS, AND ORIGINS......Page 237 §1 Introduction......Page 239 §2 Seven Theses of Intuitionism......Page 240 §3 Dummett's View of Intuitionism and Some Objections to It......Page 243 §4 Replacing the Requirement of Full Manifestability......Page 245 §5 What Are Intuitionistic Constructions?......Page 249 §6 Constructions and Manifestations of the Understanding of Meaning in Linguistic Behavior......Page 251 §7 Intentions and Constructions......Page 254 §9 Conclusion: An Alternative to Dummett's View......Page 258 12 The Philosophical Background of Weyl's Mathematical Constructivism......Page 260 §1 Weyl and Idealism......Page 261 §2 Weyl: Idealism and Epistemology in Mathematics......Page 265 §3 Intuition in Mathematics: Husserl and Weyl......Page 267 §4 Intuition, Categories, Founding, and Existence......Page 271 §5 Meaning in Mathematics......Page 273 §6 Time Consciousness, the Continuum, and Choice Sequences......Page 275 §7 The Intuitive and the Symbolic in Mathematics......Page 283 §8 Conclusion......Page 287 13 Proofs and Fulfillable Mathematical Intentions......Page 288 §1......Page 290 §2......Page 294 §3......Page 298 §4......Page 300 14 Logicism, Impredicativity, Formalism: Some Remarks on Poincaré and Husserl......Page 306 §1 Logicism......Page 307 §2 Impredicativity......Page 317 §3 Formalism......Page 322 15 The Philosophy of Arithmetic: Frege and Husserl......Page 326 §1 Frege on the Foundations of Arithmetic: Logicism......Page 327 §2 The Frege-Husserl Dispute over Arithmetic......Page 330 §3 Frege's Later Philosophy of Arithmetic......Page 334 §4 Husserl and the Philosophy of Arithmetic:What Is the Origin of Our Knowledge of Ideal Objects?......Page 337 §5 The Role of Intuition in Arithmetic Knowledge......Page 341 §6 A New View of the Foundations of Arithmetic......Page 343 §7 Conclusion......Page 347 Bibliography......Page 349 Index......Page 361 This Book Is A Collection Of Fifteen Essays That Deal With Issues At The Intersection Of Phenomenology, Logic, And The Philosophy Of Mathematics.--book Jacket. Introduction : Themes And Issues -- Pt. I. Reason, Science, And Mathematics -- 1. Science As A Triumph Of The Human Spirit And Science In Crisis : Husserl And The Fortunes Of Reason -- 2. Mathematics And Transcendental Phenomenology -- 3. Free Variation And The Intuition Of Geometric Essences : Some Reflections On Phenomenology And Modern Geometry -- Pt. Ii. Kurt Godel, Phenomenology, And The Philosophy Of Mathematics -- 4. Kurt Godel And Phenomenology -- 5. Godel's Philosophical Remarks On Logic And Mathematics -- 6. Godel's Path From The Incompleteness Theorems (1931) To Phenomenology (1961) -- 7. Godel And The Intuition Of Concepts -- 8. Godel And Quine On Meaning And Mathematics -- 9. Maddy On Realism In Mathematics -- 10. Penrose On Minds And Machines -- Pt. Iii. Constructivism, Fulfillable Intentions, And Origins -- 11. Intuitionism, Meaning Theory, And Cognition -- 12. Philosophical Background Of Weyl's Mathematical Constructivism -- 13. Proofs And Fulfillable Mathematical Intentions -- 14. Logicism, Impredicativity, Formalism : Some Remarks On Poincare And Husserl -- 15. Philosophy Of Arithmetic : Frege And Husserl. Richard Tieszen. Includes Bibliographical References (p. 337-348) And Index. Offering a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics, this book is divided into three parts. Part I, Reason, Science, and Mathematics contains a general essay on Husserl's conception of science and logic, an essay of mathematics and transcendental phenomenology, and an essay oN phenomenology and modern pure geometry. Part II is focused on Kurt Godel's interest in phenomenology. It explores Godel's ideas and also some work of Quine, Penelope Maddy and Roger Penrose. Part III deals with elementary, constructive areas of mathematics. These are areas of mathematics that are closer to their origins in simple cognitive activities and in everyday experience. This part of the book contains essays on intuitionism, Hermann Weyl, the notion of constructive proof, Poincar © and Frege. Logic, mathematical knowledge and objects are explored alongside reason and intuition in the exact sciences