Philosophy of Arithmetic

542 Pages·2013·15.84 MB·English

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
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© 2003 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2003
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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