The Origin of the Logic of Symbolic Mathematics: Edmund Husserl and Jacob Klein (Studies in Continental Thought)

Burt C. Hopkins presents the first in-depth study of the work of Edmund Husserl and Jacob Klein on the philosophical foundations of the logic of modern symbolic mathematics. Accounts of the philosophical origins of formalized concepts—especially mathematical concepts and the process of mathematical abstraction that generates them—have been paramount to the development of phenomenology. Both Husserl and Klein independently concluded that it is impossible to separate the historical origin of the thought that generates the basic concepts of mathematics from their philosophical meanings. Hopkins explores how Husserl and Klein arrived at their conclusion and its philosophical implications for the modern project of formalizing all knowledge. The Author Presents The First In-depth Study Of The Work Of Edmund Husserl And Jacob Klein On The Philosophical Foundations Of The Logic Of Modern Symbolic Mathematics. Accounts Of The Philosophical Origins Of Formalized Concepts--especially Mathematical Concepts And The Process Of Mathematical Abstraction That Generates Them--have Been Paramount To The Development Of Phenomenology. Both Husserl And Klein Independently Concluded That It Is Impossible To Separate The Historical Origin Of The Thought That Generates The Basic Concepts Of Mathematics From Their Philosophical Meanings. He Explores How Husserl And Klein Arrived At Their Conclusion And Its Philosophical Implications For The Modern Project Of Formalizing All Knowledge. Introduction: The Subject Matter, Thesis, And Structure Of This Study. Part One: Klein On Husserl's Phenomenology And The History Of Science. Chapter One: Klein's And Husserl's Investigations Of The Originiation Of Mathematical Physics -- Chapter Two: Klein's Account Of The Essential Connection Between Intentional And Actual History -- Chapter Three: The Liberation Of The Problem Of Origin From Its Naturalistic Distortion: The Phenomenological Problem Of Constitution -- Chapter Four: The Essential Connection Between Intentional And Actual History -- Chapter Five: The Historicity Of The Intelligibility Of Ideal Significations And The Possibility Of Actual History -- Chapter Six: Sedimentation And The Link Between Intentional History And The Constitution Of Historical Tradition -- Chapter Seven: Klein's Departure From The Content But Not The Method Of Husserl's Intentional-historical Analysis Of Modern Science. Part Two: Husserl And Klein On The Method And Task Of Desedimenting The Mathematization Of Nature. Chapter Eight: Klein's Historical-mathematical Investigations In The Context Of Husserl's Phenomenology Of Science -- Chapter Nine: The Basic Problem And Method Of Klein's Mathematical Investigations -- Chapter Ten: Husserl's Formulation Of The Nature And Roots Of The Crisis Of European Sciences -- Chapter Eleven: The Zigzag Movement Implicit In Klein's Mathematical Investigations -- Chapter Twelve: Husserl And Klein On The Logic Of Symbolic Mathematics. Part Three: Non-symbolic And Symbolic Numbers In Husserl And Klein. Chapter Thirteen: Authentic And Symbolic Numbers In Husserl's Philosophy Of Arithmetic -- Chapter Fourteen: Klein's Desedimentation Of The Origin Of Algebra And Husserl's Failure To Ground Symbolic Calculation In Authentic Numbers -- Chapter Fifteen: Logistic And Arithmetic In Neoplatonic Mathematics And In Plato -- Chapter Sixteen: Theoretical Logistic And The Problem Of Fractions -- Chapter Seventeen: The Concept Of Api0mog -- Chapter Eighteen: Plato's Ontological Conception Of Api0moi -- Chapter Nineteen: Klein's Reactivation Of Plato's Theory Of Api0moi Eiontikoi -- Chapter Twenty: Aristotle's Critique Of The Platonic Chorismos Thesis And The Possibility Of A Theoretical Logistic -- Chapter Twenty-one: Klein's Interpretation Of Diophantus' Arithmetic. Chapter Twenty-two: Klein's Account Of Vieta's Reinterpretation Of The Diophantine Procedure And The Consequent Establishment Of Algebra As The General Analytical Art. Chapter Twenty-three: Klein's Account Of The Concept Of Number And The Number Concepts In Stervin, Descartes, And Wallis. Part Four: Husserl And Klein On The Origination Of The Logic And Symbolic Mathematics. Chapter Twenty-four: Husserl And Klein On The Fundamental Difference Between Symbolic And Non-symbolic Numbers -- Chapter Twenty-five: Husserl And Klein On The Origin And Structure Of Non-symbolic Numbers -- Chapter Twenty-six: Structural Differences In Husserl's And Klein's Accounts Of The Mode Of Being Of Non-symbolic Numbers -- Chapter Twenty-seven: Digression: The Development Of Husserl's Though, After Philosophy Of Arithmetic, On The Logical Status Of The Symbolic Calculus, The Constitution Of Collective Unity, And At The Phenomenological Foundation Of The Mathesis Universalis -- Chapter Twenty-eight: Husserl's Accounts Of The Symbolic Calculus, The Critique Of Psychologism, And The Phenomenological Foundation Of The Mathesis Universalis After Philosophy Of Arithmetic -- Chapter Twenty-nine: Husserl's Critique Of Symbolic Calculation In His Schroder Review --^ Chapter Thirty: The Separation Of Logic From Symbolic Calculation In Husserl's Later Works -- Chapter Thirty-one: Husserl On The Shortcomings Of The Appeal To The Reflexion On Acts To Account For The Origin Of Logical Relations In The Works Leading Up To The Logical Investigations -- Chapter Thirty-two: Husserl's Attempt In The Logical Investigations To Esstablish A Relationship Between Mere Thought And The In Itself Of Pure Logical Validity By Appealing To Concrete, Universal, And Formalizing Modes Of Abstraction And Categorical Intuition. Chapter Thirty-three: Husserl's Account Of The Constitution Of The Collection, Number, And The Universal Whatever In Experience And Judgement -- Chapter Thirty-four: Husserl's Investigation Of The Unitary Domain Of Formal Logic And The Formal Ontology In Formal And Transcendental Logic -- Chapter Thirty-five: Klein And Husserl On The Origination Of The Logic Of Symbolic Numbers --^ Chapter Thirty-six: Conclusion. Glossary Of Greek And German Terms. Bibliography. Index Of Names. Index Of Subjects. Burt C. Hopkins. Includes Bibliographical References And Indexes. Cover 1 Contents 8 Preface by Eva Brann 24 Acknowledgments 30 List of Abbreviations 32 Introduction. The Subject Matter, Thesis, and Structure of This Study 36 Part One. Klein on Husserl’s Phenomenology and the History of Science 44 Chapter One. Klein’s and Husserl’s Investigations of the Originationof Mathematical Physics 46 Chapter Two. Klein’s Account of the Essential Connection between Intentional and Actual History 57 Chapter Three. The Liberation of the Problem of Origin from Its Naturalistic Distortion: The Phenomenological Problem of Constitution 62 Chapter Four. The Essential Connection between Intentional and Actual History 66 Chapter Five. The Historicity of the Intelligibility of Ideal Significations and the Possibility of Actual History 74 Chapter Six. Sedimentation and the Link between Intentional History and the Constitution of a Historical Tradition 79 Chapter Seven. Klein’s Departure from the Content but Not the Method of Husserl’s Intentional-Historical Analysis of Modern Science 88 Part Two. Husserl and Klein on the Method and Task of Desedimenting the Mathematization of Nature 98 Chapter Eight. Klein’s Historical-Mathematical Investigations in the Context of Husserl’s Phenomenology of Science 100 Chapter Nine. The Basic Problem and Method of Klein’s Mathematical Investigations 108 Chapter Ten. Husserl’s Formulation of the Nature and Roots of the Crisis of European Sciences 116 Chapter Eleven. The “Zigzag” Movement Implicit in Klein’s Mathematical Investigations 128 Chapter Twelve. Husserl and Klein on the Logic of Symbolic Mathematics 132 Part Three. Non-symbolic and Symbolic Numbers in Husserl and Klein 136 Chapter Thirteen. Authentic and Symbolic Numbers in Husserl’s Philosophy of Arithmetic 138 Chapter Fourteen. Klein’s Desedimentation of the Origin of Algebra and Husserl’s Failure to Ground Symbolic Calculation in Authentic Numbers 182 Chapter Fifteen. Logistic and Arithmetic in Neoplatonic Mathematics and in Plato 187 Chapter Sixteen. Theoretical Logistic and the Problem of Fractions 202 Chapter Seventeen. The Concept of ̓Αριθμός 208 Chapter Eighteen. Plato’s Ontological Conception of ̓Αριθμοί 218 Chapter Nineteen. Klein’s Reactivation of Plato’s Theory of ̓Αριθμοὶ Εἰδητικοί 231 Chapter Twenty. Aristotle’s Critique of the Platonic Chorismos Thesis and the Possibility of a Theoretical Logistic 259 Chapter Twenty-one. Klein’s Interpretation of Diophantus’s Arithmetic 270 Chapter Twenty-two. Klein’s Account of Vieta’s Reinterpretation of the Diophantine Procedure and the Consequent Establishment of Algebra as the General Analytical Art 287 Chapter Twenty-three. Klein’s Account of the Concept of Number and the Number Concepts in Stevin, Descartes, and Wallis 327 Part Four. Husserl and Klein on the Origination of the Logic of Symbolic Mathematics 360 Chapter Twenty-four. Husserl and Klein on the Fundamental Difference between Symbolic and Non-symbolic Numbers 362 Chapter Twenty-five. Husserl and Klein on the Origin and Structure of Non-symbolic Numbers 370 Chapter Twenty-six. Structural Differences in Husserl’s and Klein’s Accounts of the Mode of Being of Non-symbolic Numbers 387 Chapter Twenty-seven. Digression: The Development of Husserl’sThought, after Philosophy of Arithmetic, on the “Logical” Status of the Symbolic Calculus, the Constitution of Collective Unity, and the Phenomenological Foundation of the Mathesis Universalis 395 Chapter Twenty-eight. Husserl’s Accounts of the Symbolic Calculus, the Critique of Psychologism, and the Phenomenological Foundation of the Mathesis Universalis after Philosophy of Arithmetic 397 Chapter Twenty-nine. Husserl’s Critique of Symbolic Calculation in his Schröder Review 405 Chapter Thirty. The Separation of Logic from Symbolic Calculation in Husserl’s Later Works 411 Chapter Thirty-one. Husserl on the Shortcomings of the Appeal to the “Reflexion” on Acts to Account for the Origin of Logical Relations in the Works Leading up to the Logical Investigations 419 Chapter Thirty-two. Husserl’s Attempt in the Logical Investigations to Establish a Relationship between “Mere” Thought and the “In Itself ” of Pure Logical Validity by Appealing to Concrete, Universal, and Formalizing Modes of Abstraction and Categorial Intuition 425 Chapter Thirty-three. Husserl’s Account of the Constitution of the Collection, Number, and the ‘Universal Whatever’ in Experience and Judgment 445 Chapter Thirty-four. Husserl’s Investigation of the Unitary Domain of Formal Logic and Formal Ontology in Formal and Transcendental Logic 467 Chapter Thirty-five. Klein and Husserl on the Origination of the Logic of Symbolic Numbers 524 Chapter Thirty-six. Conclusion 550 Glossary of Greek and German Terms 574 Bibliography 578 Index of Names 586 Index of Subjects 588