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Philosophy of Arithmetic: Psychological and Logical Investigations with Supplementary Texts from 1887–1901 PDF

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PHILOSOPHY OF ARITHMETIC EDMUND HUSSERL COLLECTED WORKS EDITOR: RUDOLF BERNET VOLUME X PHILOSOPHY OF ARITHMETIC Psychological and Logical Investigations with Supplementary Texts from 1887-1901 TRANSLATIONS PREPARED UNDER THE AUSPICES OF THE HUSSERL-ARCHIVES (LEUVEN) A list of titles in this series can be found at the end of this volume. EDMUND HUSSERL PHILOSOPHY OF ARITHMETIC Psychological and Logical Investigations with Supplementary Texts from 1887-1901 TRANSLATED BY DALLAS WILLARD School of Philosophy, University of Southern California, Las Angeles SPRINGER SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress ISBN 978-1-4020-1603-5 ISBN 978-94-010-0060-4 (eBook) DOI 10.1007/978-94-010-0060-4 Printed an acid-free pap er AH Rights Reserved © 2003 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2003 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permis sion from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. CONTENTS TRANSLATOR'S INTRODUCTION .................................................. xiii PHILOSOPHY OF ARITHMETIC: PSYCHOLOGICAL AND LOGICAL INVESTIGATIONS Volume One FOREWORD ........................................................................................... 5 FIRST PART: THE AUTHENTIC CONCEPTS OF MULTIPLICITY, UNITY AND WHOLE NUMBER ........................... 9 INTRODUCTION .......................................................................... 11 Chapter I: THE ORIGINATION OF THE CONCEPT OF MULTIPLICITY THROUGH THAT OF THE COLLECTIVE COMBINATION ........................................................................... 15 The Analysis of the Concept of the Whole Number Presupposes that of the Concept of Multiplicity ................... 15 The Concrete Bases of the Abstraction Involved ..................... 16 Independence of the Abstraction from the Nature of the Contents Colligated ............................................................... 17 The Origination ofthe Concept of the Multiplicity through Reflexion on the Collective Mode of Combination .............. 18 Chapter II: CRITICAL DEVELOPMENTS .................................. 23 The Collective Unification and the Unification of Partial Phenomena in the Total Field of Consciousness at a Given Moment ...................................................................... 23 The Collective "Together" and the Temporal "Simultaneously" .................................................................. 25 Collection and Temporal Succession ....................................... 26 The Collective Synthesis and the Spatial Synthesis ................. 35 A. F. A. Lange's Theory ...................................................... 35 B. Baumann's Theory .......................................................... 45 VI PHILOSOPHY OF ARITHMETIC Colli gating, Enumerating and Distinguishing .......................... 49 Critical Supplement .............................................................. 61 Chapter III: THE PSYCHOLOGICAL NATURE OF THE COLLECTIVE COMBINATION ................................................. 67 Review ..................................................................................... 67 The Collection as a Special Type of Combination ................... 68 On The Theory of Relations ..................................................... 69 Psychological Characterization of the Collective Combination .......................................................................... 74 Chapter IV: ANALYSIS OF THE CONCEPT OF NUMBER IN TERMS OF ITS ORIGIN AND CONTENT ............................ 81 Completion of the Analysis of the Concept of Multiplicity ..... 81 The Concept 'Something' ........................................................ 84 The Cardinal Numbers and the Generic Concept of Number .. 85 Relationship Between the Concepts 'Cardinal Number' and 'Multiplicity' .................................................................. 87 One and Something .................................................................. 88 Critical Supplement .............................................................. 89 Chapter V: THE RELATIONS "MORE" AND "LESS" ............... 95 The Psychological Origin of These Relations .......................... 95 Comparison of Arbitrary Multiplicities, as well as of Numbers, in Terms of More and Less ................................... 98 The Segregation of the Number Species Conditioned upon the Knowledge of More and Less ................................. 99 Chapter VI: THE DEFINITION OF NUMBER-EQUALITY THROUGH THE CONCEPT OF RECIPROCAL ONE-TO- ONE CORRELATION ................................................................ 101 Leibniz's Definition of the General Concept of Equality ...... 101 The Definition of Number-Equality ....................................... 103 Concerning Definitions of Equality for Special Cases ........... 105 Application to the Equality of Arbitrary Multiplicities .......... 106 Comparison of Multiplicities of One Genus .......................... 108 Comparison of Multiplicities with Respect to their Number. 108 The True Sense of the Equality Definition under Discussion ........................................................................... 110 Reciprocal Correlation and Collective Combination ............. 111 The Independence of Number-Equality from the Type of Linkage ........................................................................... 114 CONTENTS Vll Chapter VII: DEFINITIONS OF NUMBER IN TERMS OF EQUIVALENCE ......................................................................... 117 Structure of the Equivalence Theory ...................................... 117 Illustrations ............................................................................ 120 Critique ............................................................................... 121 Frege's Attempt .............................................................. 123 Kerry's Attempt. ............................................................. 129 Concluding Remark. ....................................................... 131 Chapter VIII: DISCUSSIONS CONCERNING UNITY AND MULTIPLICITY. ............................................................... 133 The Definition of Number as a Multiplicity of Units ............. 133 One as an Abstract, Positive Partial Content... ....................... 133 One as Mere Sign ................................................................... 133 One and Zero as Numbers ...................................................... 136 The Concept of the Unit and the Concept of the Number One ........................................................................ 141 Further Distinctions Concerning One and Unit... ................... 143 Sameness and Distinctness of the Units ................................. 146 Further Misunderstandings .................................................... 157 Equivocations of the Name "Unit" ........................................ 159 The Arbitrary Character of the Distinction between Unit and Multiplicity. The Multiplicity Regarded as One Multiplicity, as One Enumerated Unit, as One Whole ....... 162 Herbartian Arguments ............................................................ 164 Chapter IX: THE SENSE OF THE STATEMENT OF NUMBER .............................................................................. 169 Contradictory Views .............................................................. 169 Refutation, and the Position Taken ........................................ 170 APPENDIX TO THE FIRST PART: The Nominalist Attempts of Helmholtz and Kronecker ................................................................... 179 SECOND PART: THE SYMBOLIC NUMBER CONCEPTS AND THE LOGICAL SOURCES OF CARDINAL ARITHMETIC ......... 189 Chapter X: OPERATIONS ON NUMBERS AND THE AUTHENTIC NUMBER CONCEPTS ....................................... 191 The Numbers in Arithmetic Are Not Abstracta ..................... 191 The Fundamental Activities on Numbers ............................... 192 Addition ................................................................................. 193 V111 PHILOSOPHY OF ARITHMETIC Partition .................................................................................. 198 Arithmetic Does Not Operate with "Authentic" Number Concepts .............................................................................. 200 Chapter XI: SYMBOLIC REPRESENTATIONS OF MULTIPLICITIES ...................................................................... 205 Authentic and Symbolic Representations .............................. 205 Sense Perceptible Groups ....................................................... 207 Attempts at an Explanation of How We Grasp Groups in an Instant ......................................................................... 208 Symbolizations Mediated by the Full Process of Apprehending the Individual Elements ............................... 210 New Attempts at an Explanation ofInstantaneous Apprehensions of Groups .................................................... 211 Hypotheses ............................................................................. 213 The Figural Moments ............................................................. 215 The Position Taken ................................................................ 223 The Psychological Function of the Focus upon Individual Members of the Group ....................................... 225 What Is It that Guarantees the Completeness of the Traversive Apprehension of the Individuals in a Group? ... 226 Apprehension of Authentically Representable Groups through Figural Moments .................................................... 228 The Elemental Operations on and Relations between Multiplicities Extended to Symbolically Represented Multiplicities ....................................................................... 229 Infinite Groups ....................................................................... 230 Chapter XII: THE SYMBOLIC REPRESENTATIONS OF NUMBERS ............................................................................ 235 The Symbolic Number Concepts and their Infinite Multiplicity .......................................................................... 235 The Non-Systematic Symbolizations of Numbers ................. 236 The Sequence of Natural Numbers ........................................ 238 The System of Numbers ......................................................... 241 Relationship of the Number System to the Sequence of Natural Numbers ............................................................. 247 The Choice of the "Base Number" for the System ................ 249 The Systematic of the Number Concepts and the Systematic of the Number Signs ......................................... 251 CONTENTS IX The Process of Enumeration via Sense Perceptible Symbols 253 Expansion of the Domain of Symbolic Numbers through Sense Perceptible Symbolization ........................... 254 Differences between Sense Perceptible Means of Designation ......................................................................... 257 The Natural Origination of the Number System .................... 258 Appraisal of Number through Figural Moments .................... 267 Chapter XIII: THE LOGICAL SOURCES OF ARITHMETIC .. 271 Calculation, Calculational Technique and Arithmetic ........... 271 The Calculational Methods of Arithmetic and the Number Concepts ................................................................ 274 The Systematic Numbers as Surrogates for the Numbers in Themselves ...................................................... 275 The Symbolic Number Formations that Fall Outside the System, Viewed as Arithmetical Problems ................... 275 The First Basic Task of Arithmetic ........................................ 277 The Elemental Arithmetical Operations ................................. 277 Addition ................................................................................. 279 Multiplication ................................................................ , ........ 283 Subtraction and Division ........................................................ 284 Methods of Calculation with the Abacus and in Columns. The Natural Origination of the Indic Numeral Calculation. 288 Influence of the Means of Designation upon the Formation of the Methods of Calculation ........................... 290 The Higher Operations ........................................................... 292 Mixing of Operations ............................................................. 294 The Indirect Characterization of Numbers by Means of Equations ........................................................................ 296 Result: The Logical Sources of General Arithmetic ............. 298 SELBSTA NZEIGE - PHILOSOPHIE DER ARITHMETIK .............. 301 SUPPLEMENTARY TEXTS (1887 - 1901) A. ORIGINAL VERSION OF THE TEXT THROUGH CHAPTER IV: ON THE CONCEPT OF NUMBER: PSYCHOLOGICAL ANALYSES ................................................................................ 305 Introduction ............................................................................ 305 Chapter One ........................................................................... 312 x PHILOSOPHY OF ARITHMETIC THE ANALYSIS OF THE CONCEPT OF NUMBER AS TO ITS ORIGIN AND CONTENT .............................. 312 Section 1: The Formation of the Concept of Multiplicity [Vielheit] out of That of the Collective Combination ........................................... 312 Section 2: Critical Exposition of Certain Theories ............ 318 Section 3: Establishment of the "Psychological" Nature of the Collective Combination .......................... 344 Section 4: The Analysis of the Concept of Number as to its Origin and Content... ........................................ 352 APPENDIX TO "ON THE CONCEPT OF NUMBER: PSYCHOLOGICAL ANALYSES" - THESES .......................... 357 B. ESSAYS .......................................................................................... 359 ESSAY I: (ON THE THEORY OF THE TOTALITY) ........ 359 (I. The Definition of the Totality) ....................................... 359 (II. Comparison of Numbers ) .............................................. 364 (III. Addenda ) ...................................................................... 368 ( 1. Addendum to p. 367: Identity and Equality) ................. 368 (2. On the Definition of Number ) ........................................ 369 (IV. The Classification of the Cardinal Numbers) .............. 369 (V. Remark ) ......................................................................... 374 (VI. Corrections) ................................................................. 375 (VII. Addenda) .................................................................... 379 (1. Addendum to p. 369 ) .................................................. 379 (2. Addendumtop.377) .................................................. 380 ESSAY II: (ON THE CONCEPT OF THE OPERATION) .. 385 (I. Arithmetical Determinations of Number ) ....................... 385 ( II. Combinations (or Operations) ) ....................................... 397 ( 1. Division ) ...................................................................... 397 (2. On the Concept of Combination [Verkniipfung] ) ...... .400 (III. Addendum) .................................................................. 405 On the Concept of Basic Operation ....................................... 405 ESSA Y III: (DOUBLE LECTURE: ON THE TRANSITION THROUGH THE IMPOSSIBLE ("IMAGINARY") AND THE COMPLETENESS OF AN AXIOM SYSTEM) .................................................... .409 (I. For a Lecture before the Mathematical Society of G6ttingen 1901 ) ............................................................. 409

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In his first book, Philosophy of Arithmetic, Edmund Husserl provides a carefully worked out account of number as a categorial or formal feature of the objective world, and of arithmetic as a symbolic technique for mastering the infinite field of numbers for knowledge. It is a realist account of numb
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